math 31 lessons chapters 6 & 7: trigonometry 9. derivatives of other trig functions

MATH 31 LESSONS

Chapters 6 & 7:

Trigonometry

9. Derivatives of Other Trig Functions

Section 7.3: Derivatives of Other

Trigonometric Functions

Read Textbook pp. 315 - 319

Derivatives of Other Trig Functions

We will now look at the derivatives of other

trig functions.

Each of these formulas are derived from the

derivatives of sine / cosine, as will be shown

in the first example.

Formulas:

xxdx

d 2sectan

xxdx

d 2csccot

xxxdx

dcotcsccsc

xxxdx

dtansecsec

Note: If the trig functions have an “inside function u(x)”,

then the formulas can be altered as follows:

dx

duuu

dx

d 2sectan

dx

duuu

dx

d 2csccot

dx

duuuu

dx

d cotcsccsc

dx

duuuu

dx

d tansecsec

Ex. 1 Prove that

Try this example on your own first.Then, check out the solution.

xxdx

d 2csccot

x

x

dx

dx

dx

d

sin

coscot

Reduce the trig function to sine and cosine.

x

x

dx

dx

dx

d

sin

coscot

2sin

sincossincos

x

xxxx

2v

vuvu

v

u

Quotient Rule:

x

x

dx

dx

dx

d

sin

coscot

2sin

sincossincos

x

xxxx

2sin

coscossinsin

x

xxxx

2sin

coscossinsin

x

xxxx

2

22

sin

cossin

x

xx

2

22

sin

cossin

x

xx

Simplify and factor out the negative from the numerator.

2sin

coscossinsin

x

xxxx

2

22

sin

cossin

x

xx

2

22

sin

cossin

x

xx

x2sin

1

x2csc 1cossin 22

Ex. 2 Differentiate.

Try this example on your own first.Then, check out the solution.

3

4

1sec12 xy

3

4

1sec12 xy

3

4

1Let xxuu

uy sec12

If it helps, substitute u(x) for the “inside function”

uy sec12

dx

duuuy tansec12

Find the derivative of the “outside function”.

Leave the inside function the same.

Don’t forget to find the derivative of the “inside function”

dx

duuuy tansec12

3

4

1Let xxuu

333

4

1

4

1tan

4

1sec12 x

dx

dxxy

Back substitute.

333

4

1

4

1tan

4

1sec12 x

dx

dxxy

233 3

4

1

4

1tan

4

1sec12 xxx

332

4

1tan

4

1sec9 xxx

Ex. 3 Differentiate.

Try this example on your own first.Then, check out the solution.

xy costan2

xy costan2

2costan xy

Remember, the entire tangent function is being squared.

i.e. tan 2 x = (tan x)2

xy costan2

2costan xy

xdx

dxy costancostan2

Find the derivative of the “outside function”.

Leave the inside function the same.

Don’t forget to find the derivative of the “inside function”

Chain rule

xdx

dxy costancostan2

xdx

dxx coscosseccostan2 2

Chain rule again.

xdx

dxy costancostan2

xdx

dxx coscosseccostan2 2

xxx sincosseccostan2 2

xxx costancossecsin2 2

Ex. 4 Differentiate.

Try this example on your own first.Then, check out the solution.

12csc

1

xy

2

112csc

12csc

1

xx

y

First, write in exponential notation.

2

112csc

12csc

1

xx

y

12csc12csc2

12

3 x

dx

dxy

Find the derivative of the “outside function”.

Leave the inside function the same.

Don’t forget to find the derivative of the “inside function”

Chain rule

2

112csc

12csc

1

xx

y

12csc12csc2

12

3 x

dx

dxy

1212cot12csc12csc2

12

3 x

dx

dxxx

Chain rule again.

1212cot12csc12csc

2

12

3 x

dx

dxxx

212cot12csc12csc2

12

3 xxx

1212cot12csc12csc

2

12

3 x

dx

dxxx

212cot12csc12csc2

12

3 xxx

12cot12csc12csc22

12

3 xxx

Bring all the coefficients together

1212cot12csc12csc

2

12

3 x

dx

dxxx

212cot12csc12csc2

12

3 xxx

12cot12csc12csc 23

xxx

12cot12csc12csc22

12

3 xxx

12cot12csc12csc 23

xxx

12cot12csc 21

xx

Notice,

u3/2 u1 = u1/2

12cot12csc12csc 23

xxx

12cot12csc 21

xx

12csc

12cot

x

x