math 112 lecture 13: derivatives of trigonometric functionspeople.okanagan.bc.ca › clee ›...

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Lecture 13: Derivatives of Trigonometric Functions

Derivatives of the Basic Trigonometric FunctionsDerivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions

Applying the Trig Function Derivative RulesExample 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations

Derivatives of the Inverse Trigonometric FunctionsThe arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 1/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

The sum and difference formulas for sin

sin (x ± y)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

The sum and difference formulas for sin

sin (x ± y) = sin x cos y ± cos x sin y

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

The sum and difference formulas for sin

sin (x ± y) = sin x cos y ± cos x sin y

The corresponding formula for cos

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

The sum and difference formulas for sin

sin (x ± y) = sin x cos y ± cos x sin y

The corresponding formula for cos

cos (x ± y)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Two Identities

In Example 31(c) we guessed that

d

dxsin x = cos x

from the graphs of sin and cos. This can be proved using the limitdefinition of the derivative together with some basic identities.

Sum and Difference Formulas for sin and cos

The sum and difference formulas for sin

sin (x ± y) = sin x cos y ± cos x sin y

The corresponding formula for cos

cos (x ± y) = cos x cos y ∓ sin x sin y

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 2/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

= limh→0

sin x cos h + cos x sin h − sin x

h

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

= limh→0

sin x cos h + cos x sin h − sin x

h

= limh→0

sin x (cos h − 1) + cos x sin h

h

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

= limh→0

sin x cos h + cos x sin h − sin x

h

= limh→0

sin x (cos h − 1) + cos x sin h

h

= sin x

(

limh→0

cos h − 1

h

)

+ cos x

(

limh→0

sin h

h

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

= limh→0

sin x cos h + cos x sin h − sin x

h

= limh→0

sin x (cos h − 1) + cos x sin h

h

= sin x

(

limh→0

cos h − 1

h

)

+ cos x

(

limh→0

sin h

h

)

Earlier we used the Squeeze Theorem to proved that

limx→0

sin x

x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin

With f (x) = sin x, using Formula 3 we have

f ′(x) = limh→0

sin (x + h) − sin x

h

= limh→0

sin x cos h + cos x sin h − sin x

h

= limh→0

sin x (cos h − 1) + cos x sin h

h

= sin x

(

limh→0

cos h − 1

h

)

+ cos x

(

limh→0

sin h

h

)

Earlier we used the Squeeze Theorem to proved that

limx→0

sin x

x= 1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 3/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Thus

f ′(x) =d

dxsin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Thus

f ′(x) =d

dxsin x = sin x (0) + cos x (1)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Thus

f ′(x) =d

dxsin x = sin x (0) + cos x (1) = cos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Thus

f ′(x) =d

dxsin x = sin x (0) + cos x (1) = cos x

You use the sum formula for cos to prove the correspondingdifferentiation formula for cos x, which is

d

dxcos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of sin

Derivative of sin – continued

As in Example 13 we can show that

limh→0

cos h − 1

h= 0

Thus

f ′(x) =d

dxsin x = sin x (0) + cos x (1) = cos x

You use the sum formula for cos to prove the correspondingdifferentiation formula for cos x, which is

d

dxcos x = − sin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 4/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

=

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Using the Chain Rule on the first of the identities above

d

dxcos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Using the Chain Rule on the first of the identities above

d

dxcos x =

d

dxsin(

π

2 − x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Using the Chain Rule on the first of the identities above

d

dxcos x =

d

dxsin(

π

2 − x)

= cos(

π

2 − x) d

dx

(

π

2 − x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Using the Chain Rule on the first of the identities above

d

dxcos x =

d

dxsin(

π

2 − x)

= cos(

π

2 − x) d

dx

(

π

2 − x)

= sin x(−1)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Derivative of cos Using the Chain Rule

Complementary Angle Identities

sin(

π

2 − x)

= cos x

cos(

π

2 − x)

= sin x

since x and π

2 − x are complementary angles in a right triangle.

Using the Chain Rule on the first of the identities above

d

dxcos x =

d

dxsin(

π

2 − x)

= cos(

π

2 − x) d

dx

(

π

2 − x)

= sin x(−1) = − sin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 5/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Two Basic Derivative Formulas

Derivatives of sin and cos

d

dxsin u =

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 6/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Two Basic Derivative Formulas

Derivatives of sin and cos

d

dxsin u = cos u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 6/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Two Basic Derivative Formulas

Derivatives of sin and cos

d

dxsin u = cos u

du

dxd

dxcos u =

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 6/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Two Basic Derivative Formulas

Derivatives of sin and cos

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 6/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of cos Using the Chain Rule

Two Basic Derivative Formulas

Derivatives of sin and cos

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dx

Here the Chain Rule form of the derivative formulas are given, inwhich the variable u is a function of x.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 6/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

=(cos x) cos x − sin x (− sin x)

cos2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

=(cos x) cos x − sin x (− sin x)

cos2 x

=cos2 x + sin2 x

cos2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

=(cos x) cos x − sin x (− sin x)

cos2 x

=cos2 x + sin2 x

cos2 x=

1

cos2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

=(cos x) cos x − sin x (− sin x)

cos2 x

=cos2 x + sin2 x

cos2 x=

1

cos2 x=

(

1

cos x

)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule

Recall that

tan x =sin x

cos x

Thus, using the Quotient Rule gives

d

dxtan x =

d

dx

sin x

cos x

=

(

d

dxsin x

)

cos x − sin x

(

d

dxcos x

)

(cos x)2

=(cos x) cos x − sin x (− sin x)

cos2 x

=cos2 x + sin2 x

cos2 x=

1

cos2 x=

(

1

cos x

)2

= sec2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 7/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x = sec2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x = sec2 x = 1 + tan2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x = sec2 x = 1 + tan2 x

The equality above can also be proved using the Pythagorean identity

1 + tan2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x = sec2 x = 1 + tan2 x

The equality above can also be proved using the Pythagorean identity

1 + tan2 x = sec2 x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivative of tan Using the Quotient Rule

Derivative of tan Using the Quotient Rule – continued

An alternative way to simplify the previous expression is

d

dxtan x =

cos2 x + sin2 x

cos2 x=

cos2 x

cos2 x+

sin2 x

cos2 x

= 1 +

(

sin x

cos x

)2

= 1 + tan2 x

So thatd

dxtan x = sec2 x = 1 + tan2 x

The equality above can also be proved using the Pythagorean identity

1 + tan2 x = sec2 x

Most text books use the sec2 x formula for the derivative of tan x, butMaple and other symbolic differentiating programs use the 1 + tan2 xformula.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 8/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u = sec u tan u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u = sec u tan u

du

dxd

dxcsc u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u = sec u tan u

du

dxd

dxcsc u = − csc u cot u

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u = sec u tan u

du

dxd

dxcsc u = − csc u cot u

du

dxd

dxcot u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives the Six Trigonometric Functions

Derivatives Formulas for the Six Trigonometric Functions

d

dxsin u = cos u

du

dxd

dxcos u = − sin u

du

dxd

dxtan u = sec2 u

du

dx=(

1 + tan2 u) du

dxd

dxsec u = sec u tan u

du

dxd

dxcsc u = − csc u cot u

du

dxd

dxcot u = − csc2 u

du

dx= −

(

1 + cot2 u) du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 9/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Example 46 – Differentiating with Trig Functions

Find and simplify the indicated derivative(s) of each function.

(a) Find f ′(x) and f ′′(x) for f (x) = x2 cos (3x).

(b) Findds

dtfor s =

cos t

sin t + cos t.

(c) Find C′(x) for C(x) = tan(

e√

1+x2)

.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 10/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

= 2x cos (3x) − 3x2 sin (3x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

= 2x cos (3x) − 3x2 sin (3x)

= x [2 cos (3x) − 3x sin (3x)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

= 2x cos (3x) − 3x2 sin (3x)

= x [2 cos (3x) − 3x sin (3x)]

For f ′′(x) use the expression in the second line. Again using theProduct and Chain Rules gives

f ′′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

= 2x cos (3x) − 3x2 sin (3x)

= x [2 cos (3x) − 3x sin (3x)]

For f ′′(x) use the expression in the second line. Again using theProduct and Chain Rules gives

f ′′(x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(a)

Using the Product Rule followed by the Chain Rule (for cos (3x))gives

f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]

= 2x cos (3x) − 3x2 sin (3x)

= x [2 cos (3x) − 3x sin (3x)]

For f ′′(x) use the expression in the second line. Again using theProduct and Chain Rules gives

f ′′(x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x)

=(

2 − 9x2)

cos (3x) − 12x sin (3x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 11/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt=

− sin t (sin t + cos t) − cos t (cos t − sin t)

(sin t + cos t)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt=

− sin t (sin t + cos t) − cos t (cos t − sin t)

(sin t + cos t)2

=− sin2 t − sin t cos t − cos2 t + cos t sin t

(sin t + cos t)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt=

− sin t (sin t + cos t) − cos t (cos t − sin t)

(sin t + cos t)2

=− sin2 t − sin t cos t − cos2 t + cos t sin t

(sin t + cos t)2

=−(

sin2 t + cos2 t)

(sin t + cos t)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt=

− sin t (sin t + cos t) − cos t (cos t − sin t)

(sin t + cos t)2

=− sin2 t − sin t cos t − cos2 t + cos t sin t

(sin t + cos t)2

=−(

sin2 t + cos2 t)

(sin t + cos t)2

= − 1

(sin t + cos t)2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(b)

Using the Quotient Rule gives

ds

dt=

− sin t (sin t + cos t) − cos t (cos t − sin t)

(sin t + cos t)2

=− sin2 t − sin t cos t − cos2 t + cos t sin t

(sin t + cos t)2

=−(

sin2 t + cos2 t)

(sin t + cos t)2

= − 1

(sin t + cos t)2

This example illustrates the fact that when simplifying derivativesinvolving trig functions, you sometimes need to use standardtrigonometric identities.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 12/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x,

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex,

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2,

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x) = 1 + x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x) = 1 + x2

Using the Chain Rule three times gives

C′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x) = 1 + x2

Using the Chain Rule three times gives

C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x) = 1 + x2

Using the Chain Rule three times gives

C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)

= sec2(

e√

1+x2) (

e√

1+x2) (

12

) (

1 + x2)−1/2

(2x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 46 – Differentiating with Trig Functions

Solution: Example 46(c)

This is a composite function with

C(x) = f (g (h (k(x))))

where

f (x) = tan x, g(x) = ex, h(x) =√

x = x1/2, k(x) = 1 + x2

Using the Chain Rule three times gives

C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)

= sec2(

e√

1+x2) (

e√

1+x2) (

12

) (

1 + x2)−1/2

(2x)

=xe

√1+x2

sec2(

e√

1+x2)

√1 + x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 13/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Example 47 – Damped Oscillations

Consider the function

q(t) = e−7t sin (24t)

This function describes damped simple harmonic motion. It givesthe position of a mass attached to a spring relative to the equilibrium(resting) position of the spring. A frictional force acts to graduallyslow the mass.

(a) Find q′(t) and q′′(t) and explain their meaning in terms of thedamped oscillatory motion.

(b) Note that q(0) = 0. This means that the initial position of themass is at the equilibrium position of the spring. Find theinitial velocity of the mass. Also find the velocity when themass first returns to the equilibrium position.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 14/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Continuing Example 47

(c) Draw a graph of the function q(t).

(d) Find the first two times when the oscillating mass turns around.Show the corresponding points on the graph of q(t).

(e) Show that the function q(t) satisfies the differential equation

d2q

dt2+ 14

dq

dt+ 625q = 0

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 15/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a)

Using the Product and Chain Rules gives

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 16/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a)

Using the Product and Chain Rules gives

q′(t)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 16/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a)

Using the Product and Chain Rules gives

q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 16/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a)

Using the Product and Chain Rules gives

q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]

= e−7t [24 cos (24t) − 7 sin (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 16/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a)

Using the Product and Chain Rules gives

q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]

= e−7t [24 cos (24t) − 7 sin (24t)]

This is the velocity of the mass at time t.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 16/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a) – continued

Taking the derivative of the expression above for q′(t) gives

q′′(t)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 17/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a) – continued

Taking the derivative of the expression above for q′(t) gives

q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] +

e−7t[

−242 sin (24t) − 7(24) cos (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 17/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a) – continued

Taking the derivative of the expression above for q′(t) gives

q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] +

e−7t[

−242 sin (24t) − 7(24) cos (24t)]

= −e−7t[

14(24) cos (24t) +(

242 − 72)

sin (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 17/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a) – continued

Taking the derivative of the expression above for q′(t) gives

q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] +

e−7t[

−242 sin (24t) − 7(24) cos (24t)]

= −e−7t[

14(24) cos (24t) +(

242 − 72)

sin (24t)]

= −e−7t [336 cos (24t) + 527 sin (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 17/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(a) – continued

Taking the derivative of the expression above for q′(t) gives

q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] +

e−7t[

−242 sin (24t) − 7(24) cos (24t)]

= −e−7t[

14(24) cos (24t) +(

242 − 72)

sin (24t)]

= −e−7t [336 cos (24t) + 527 sin (24t)]

This gives the acceleration of the mass at time t.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 17/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

The velocity at this time is

v(

π

24

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

The velocity at this time is

v(

π

24

)

= e−7π/24 [24 cos (π) − 7 sin (π)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

The velocity at this time is

v(

π

24

)

= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

The velocity at this time is

v(

π

24

)

= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24 = −9.6

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(b)

Substituting t = 0 into the expression for the velocity v(t) = q′(t)gives

v(0) = e0 [24 cos (0) − 7 sin (0)] = 24

The mass returns to the equilibrium position when q(t) = 0. The firsttime after t = 0 when this happens is when

24t = π ⇒ t =π

24

The velocity at this time is

v(

π

24

)

= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24 = −9.6

This velocity is less than the initial velocity and in the oppositedirection.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 18/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(c)

The graph of the function q(t) looks likethis.The amplitude of the motion decreasesfollowing an envelope given by thedecaying exponential function e−7t, asshown.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 19/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(c)

The graph of the function q(t) looks likethis.The amplitude of the motion decreasesfollowing an envelope given by thedecaying exponential function e−7t, asshown. As we saw in the last part, notonly does the amplitude decrease, but sodoes the velocity.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 19/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(c)

The graph of the function q(t) looks likethis.The amplitude of the motion decreasesfollowing an envelope given by thedecaying exponential function e−7t, asshown. As we saw in the last part, notonly does the amplitude decrease, but sodoes the velocity.Further, note that where the graph isconcave down, q′′(t) < 0, the mass isdecelerating. The velocity is getting lesspositive or more negative.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 19/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(c)

The graph of the function q(t) looks likethis.The amplitude of the motion decreasesfollowing an envelope given by thedecaying exponential function e−7t, asshown. As we saw in the last part, notonly does the amplitude decrease, but sodoes the velocity.Further, note that where the graph isconcave down, q′′(t) < 0, the mass isdecelerating. The velocity is getting lesspositive or more negative. And, wherethe graph is concave up, q′′(t) > 0, themass is accelerating. The velocity isgetting more positive or less negative.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 19/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7 0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7

⇒24t = arctan

(

24

7

)

+ kπ

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7

⇒24t = arctan

(

24

7

)

+ kπ

where k is any integer.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7

⇒24t = arctan

(

24

7

)

+ kπ

where k is any integer. The first twopositive t values are t1 = 0.0536 andt2 = 0.1845.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7

⇒24t = arctan

(

24

7

)

+ kπ

where k is any integer. The first twopositive t values are t1 = 0.0536 andt2 = 0.1845. At these values the tangentline to the graph of q(t) is horizontal, asshown.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(d)

The mass turns around whenv(t) = q′(t) = 0. This happens when

24 cos (24t) − 7 sin (24t) = 0

⇒ tan (24t) =24

7

⇒24t = arctan

(

24

7

)

+ kπ

where k is any integer. The first twopositive t values are t1 = 0.0536 andt2 = 0.1845. At these values the tangentline to the graph of q(t) is horizontal, asshown.

0.1 0.2 0.3 0.4

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 20/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(e)

From part (a) we have

d2q

dt2+ 14

dq

dt+ 625q

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 21/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(e)

From part (a) we have

d2q

dt2+ 14

dq

dt+ 625q

= −e−7t [336 cos (24t) + 527 sin (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 21/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(e)

From part (a) we have

d2q

dt2+ 14

dq

dt+ 625q

= −e−7t [336 cos (24t) + 527 sin (24t)]

+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 21/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(e)

From part (a) we have

d2q

dt2+ 14

dq

dt+ 625q

= −e−7t [336 cos (24t) + 527 sin (24t)]

+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)

= e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)]

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 21/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 47 – Damped Oscillations

Solution: Example 47(e)

From part (a) we have

d2q

dt2+ 14

dq

dt+ 625q

= −e−7t [336 cos (24t) + 527 sin (24t)]

+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)

= e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)]

= 0

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 21/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one

π

2−π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one, and so doesnot have an inverse function.

π

2−π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the tangentfunction,

π

2−π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the tangentfunction, as shown in the graph, to theinterval

− π

2< x <

π

2

π

2−π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the tangentfunction, as shown in the graph, to theinterval

− π

2< x <

π

2

gives a one-to-one function.

π

2−π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The arctan Function

Recall that the graph of the tangentfunction looks like this. The tangentfunction is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the tangentfunction, as shown in the graph, to theinterval

− π

2< x <

π

2

gives a one-to-one function.

π

2−π

2

So the inverse of the tangent function is defined as

arctan x = tan−1 x = the angle between −π

2 and π

2 whose tangent is x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 22/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x) =

(

1 + tan2 (arctan x)) d

dxarctan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x) =

(

1 + tan2 (arctan x)) d

dxarctan x

=(

1 + x2) d

dxarctan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x) =

(

1 + tan2 (arctan x)) d

dxarctan x

=(

1 + x2) d

dxarctan x =

d

dxx = 1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x) =

(

1 + tan2 (arctan x)) d

dxarctan x

=(

1 + x2) d

dxarctan x =

d

dxx = 1

Solving ford

dxarctan x gives

d

dxarctan x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arctan Function

The Derivative of the arctan Function

With this definition of the inverse tangent function, its domain is allreal numbers and its range is

(

−π

2 , π

2

)

. Further, we have

tan (arctan x) = x and arctan (tan x) = x for −π

2 < x <π

2

Taking the derivative of the first of the formulas above, as we did tofind the derivative of the ln function, gives

d

dxtan (arctan x) =

(

1 + tan2 (arctan x)) d

dxarctan x

=(

1 + x2) d

dxarctan x =

d

dxx = 1

Solving ford

dxarctan x gives

d

dxarctan x =

1

1 + x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 23/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one π

2−π

2

1

−1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function.

π

2−π

2

1

−1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function,

π

2−π

2

1

−1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

π

2−π

2

1

−1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

gives a one-to-one function.

π

2−π

2

1

−1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

gives a one-to-one function.

π

2−π

2

1

−1

So the inverse of the sine function is defined as

arcsin x = sin−1 x = the angle between −π

2 and π

2 whose sin is x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

gives a one-to-one function.

π

2−π

2

1

−1

So the inverse of the sine function is defined as

arcsin x = sin−1 x = the angle between −π

2 and π

2 whose sin is x

The domain of inverse sine function, so defined, is

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

gives a one-to-one function.

π

2−π

2

1

−1

So the inverse of the sine function is defined as

arcsin x = sin−1 x = the angle between −π

2 and π

2 whose sin is x

The domain of inverse sine function, so defined, is [−1, 1] and itsrange is

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The arcsin Function

Recall that the graph of the sine functionlooks like this. From this graph werealize that the sine function is notone-to-one, and so does not have aninverse function. However, restricting thedomain of the sine function, as shown inthe graph, to the interval

− π

2≤ x ≤ π

2

gives a one-to-one function.

π

2−π

2

1

−1

So the inverse of the sine function is defined as

arcsin x = sin−1 x = the angle between −π

2 and π

2 whose sin is x

The domain of inverse sine function, so defined, is [−1, 1] and itsrange is

[

−π

2 , π

2

]

.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 24/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.So that

d

dxsin (arcsin x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.So that

d

dxsin (arcsin x) =

√

1 − sin2 (arcsin x)d

dxarcsin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.So that

d

dxsin (arcsin x) =

√

1 − sin2 (arcsin x)d

dxarcsin x =

√

1 − x2d

dxarcsin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.So that

d

dxsin (arcsin x) =

√

1 − sin2 (arcsin x)d

dxarcsin x =

√

1 − x2d

dxarcsin x

Solving ford

dxarcsin x gives

d

dxarcsin x

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

The Derivative of the arcsin Function

As with the arctan function, we have

sin (arcsin x) = x and arcsin (sin x) = x for −π

2 ≤ x ≤ π

2

Taking the derivative of the first of the formulas above, as we just didfor the arctan, gives

d

dxsin (arcsin x) = cos (arcsin x)

d

dxarcsin x =

d

dxx = 1

Now for −π

2 ≤ x ≤ π

2 we have cos x ≥ 0 , and using the basic

Pythagorean identity cos2 x + sin2 x = 1, gives cos x =√

1 − sin2 x.So that

d

dxsin (arcsin x) =

√

1 − sin2 (arcsin x)d

dxarcsin x =

√

1 − x2d

dxarcsin x

Solving ford

dxarcsin x gives

d

dxarcsin x =

1√1 − x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 25/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

Derivatives of the Inverse Trigonometric Functions

Derivative Formulas for the Two Inverse Trigonometric Functions

d

dxarctan u =

1

1 + u2

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 26/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

Derivatives of the Inverse Trigonometric Functions

Derivative Formulas for the Two Inverse Trigonometric Functions

d

dxarctan u =

1

1 + u2

du

dxd

dxarcsin u

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 26/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

The arcsin Function

Derivatives of the Inverse Trigonometric Functions

Derivative Formulas for the Two Inverse Trigonometric Functions

d

dxarctan u =

1

1 + u2

du

dxd

dxarcsin u =

1√1 − u2

du

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 26/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Example 48 – Differentiating with Inverse Trig Functions

Find and simplify the indicated derivative(s) of each function.

(a) Find f ′(x) and f ′′(x) for f (x) =(

1 + x2)

arctan x.

(b) Finddy

dxfor y = arcsin

(√1 − x2

)

.

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 27/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x) = 2x arctan x +(

1 + x2)

(

1

1 + x2

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x) = 2x arctan x +(

1 + x2)

(

1

1 + x2

)

= 2x arctan x + 1

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x) = 2x arctan x +(

1 + x2)

(

1

1 + x2

)

= 2x arctan x + 1

Taking the derivative of the expression above, using the Product Ruleagain, gives

f ′′(x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x) = 2x arctan x +(

1 + x2)

(

1

1 + x2

)

= 2x arctan x + 1

Taking the derivative of the expression above, using the Product Ruleagain, gives

f ′′(x) = 2 arctan x + 2x

(

1

1 + x2

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(a)

Using the Product Rule gives

f ′(x) = 2x arctan x +(

1 + x2)

(

1

1 + x2

)

= 2x arctan x + 1

Taking the derivative of the expression above, using the Product Ruleagain, gives

f ′′(x) = 2 arctan x + 2x

(

1

1 + x2

)

= 2 arctan x +2x

1 + x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 28/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b)

Using the Chain Rule gives

dy

dx

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 29/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b)

Using the Chain Rule gives

dy

dx=

1√

1 −(√

1 − x2)2

(

1

2

)

(

1 − x2)−1/2

(−2x)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 29/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b)

Using the Chain Rule gives

dy

dx=

1√

1 −(√

1 − x2)2

(

1

2

)

(

1 − x2)−1/2

(−2x)

=

(

1√

1 − (1 − x2)

)

( −x√1 − x2

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 29/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b)

Using the Chain Rule gives

dy

dx=

1√

1 −(√

1 − x2)2

(

1

2

)

(

1 − x2)−1/2

(−2x)

=

(

1√

1 − (1 − x2)

)

( −x√1 − x2

)

=

(

1√x2

)( −x√1 − x2

)

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 29/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b)

Using the Chain Rule gives

dy

dx=

1√

1 −(√

1 − x2)2

(

1

2

)

(

1 − x2)−1/2

(−2x)

=

(

1√

1 − (1 − x2)

)

( −x√1 − x2

)

=

(

1√x2

)( −x√1 − x2

)

= − x

|x|√

1 − x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 29/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b) – continued

If 0 ≤ x < 1d

dxarcsin

(√

1 − x2)

= − 1√1 − x2

You can show that

d

dxarccos x = − 1√

1 − x2

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 30/30

Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions

Example 48 – Differentiating with Inverse Trig Functions

Solution: Example 48(b) – continued

If 0 ≤ x < 1d

dxarcsin

(√

1 − x2)

= − 1√1 − x2

You can show that

d

dxarccos x = − 1√

1 − x2

So is it true that arccos x = arcsin(√

1 − x2)

?

Clint Lee Math 112 Lecture 13: Derivatives of Trigonometric Functions 30/30