chap 2 - differentiation

7/27/2019 Chap 2 - Differentiation

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CHAPTER 2

DIFFERENTIATION

2.1 Differentiation of hyperbolic functions

2.2 Differentiation of inverse trigonometric functions

2.3 Differentiation of inverse hyperbolic functions

*Recall: Methods of differentiation

-Chain rule-Product differentiation

-Quotient differentiation

-Implicit differentiation


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2.1 Differentiation of Hyperbolic Functions

Recall: Definition:

cosh2

x -xe ex

sinh2

x -x

e ex

sinhtanh

cosh

x -x

x -x

x e ex

x e e

1 coshcoth

tanh sinh

xx

x x

1cosech =

sinhx

x

1sech

coshx

x


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Derivatives of hyperbolic functions

Example 2.1: Find the derivatives of

(a) sinh x (b) cosh x (c) tanh x

Solution:

(a)

2sinh

xx ee

dx

dx

dx

d

xee xx cosh)(2

1

(b)

2cosh

xx ee

dx

dx

dx

d

xee xx sinh)(2

1


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(c)

xx

xx

ee

ee

dx

d

xdx

d

tanh

Using quotient diff:

2xx

xxxxxxxx

ee

eeeeeeee

2

2

2

2222

24

22

xxxx

xx

xxxx

eeee

ee

eeee

xx

2

2

sechcosh

1

Using the same methods, we can obtain the derivatives

of the other hyperbolic functions and these gives us the

standard derivatives.


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Standard Derivatives

( )y f x ( )dy f xdx

cosh x sinh x

sinh x cosh x

tanh x 2sech x

sech x sech x tanh x

cosech x cosech x coth x

coth x 2cosech x


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Example 2.2:

1.Find the derivatives of the following functions:

a)

b)

c)

2. Find the derivatives of the following functions:

2

3 2

(a) cosh 3 (b) sinh(2 1)

(c) ( ) ( 1) sech (d) tanh(ln )

y x r t

g x x x y x

3.(Implicit differentiation)

Finddy

dxfrom the following expressions:

2(a) sinh 4 cosh

(b) tanh ( )

x y x y

y x y


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2.2 Differentiation Involving Inverse Trigonometric

Functions

Recall: Definition of inverse trigonometric functions

Function Domain Range

1sin x 1 1x

2 2y

1cos x 1 1x 0 y

1tan x x

2 2y

1sec x 1x 02 2

y y

1cot x x 0 y

1cosec x 1x 0 02 2

y y


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Derivatives of Inverse Trigonometric Functions

Standard Derivatives:

1.2

1

1

1)(sin

xx

dx

d

2. 2

1

1

1

)(cos xxdx

d

3.2

1

1

1)(tan

xx

dx

d

4.2

1

1

1)(cot

xx

dx

d

5.

1

1)(sec

2

1

xx

x

dx

d

6.

1

1)(csc

2

1

xxx

dx

d


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2.2.1 Derivatives of xy 1sin . (proof)

Recall: yxxy sinsin 1

for ]1,1[x and ]2,2[ y .

Because the sine function is differentiable on

]2,2[ , the inverse function is also differentiable.

To find its derivative we proceed implicitly:

Given xy sin . Differentiating w.r.t. x:

)()(sin xdx

dy

dx

d

ydx

dy

dx

dyy

cos

1

1cos

Since22

y , 0cos y , so

22 1

1

sin1

1

cos

1

xyydx

dy


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Example 2.3:

1. Differentiate each of the following functions.

(a) xxf 1tan)(

(b) )1(sin)( 1 ttg

(c) xexh 21sec)(

2. Find the derivative of:

(a) 421 )(tan xy

(b) )4ln(sin)( 1 xxf

3. Find the derivative of1 2tan (tan(3 1))y t .

4. Find the derivativedx

dyif

(a) yxyx

21tan

(b) yxy 11 cos2

)(sin


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Summary

Ifu is a differentiable function ofx, then

1.dx

du

uu

dx

d

2

1

1

1)(sin

2. dx

du

uudx

d

2

1

1

1

)(cos

3.dx

du

uu

dx

d

2

1

1

1)(tan

4.dx

du

uu

dx

d

2

1

1

1)(cot

5.

dx

du

uu

u

dx

d

1

1)(sec

2

1

6.dx

du

uuu

dx

d

1

1)(csc

2

1


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2.3 Derivatives of Inverse hyperbolic Functions

Recall: Inverse Hyperbolic Functions

Function Domain Range

1sinhy x ),( ),(

1coshy x ),1[ ),0[

1tanhy x (1, 1) ),(

xy 1coth ),1()1,( ),0()0,(

y = sech1x (0, 1] ),0[

y = cosech1x ),0()0,( ),0()0,(

Function Logarithmic form

1sinhy x

2ln 1x x

1coshy x 2ln 1x x 1tanhy x 1 1

ln ; 12 1

xx

x


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2.3.1 Proof: (2

1

1

1)(sinh

xx

dx

d

)

Recall: 1sinhy x sinhx y

To find its derivative we proceed implicitly:

Given yx sinh . Differentiating w.r.t. x:

)(sinh)( ydx

dx

dx

d

ydx

dy

dx

dyy

cosh

1

cosh1

Since y , 0cosh y , so using the identity

2 2cosh sinh 1y y :

22 1

1

sin1

1

cosh

1

xyydx

dy

2

1

1

1)(sinh

x

xdx

d


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Other ways to obtain the derivatives are:

(a)1sinhy x sinhx y then

2

y ye ex

. Hence, find dy

dx.

(b)1sinhy x =

2ln 1x x .

Hence, finddy

dx.


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Standard Derivatives

Function, y Derivatives, dydx

1sinh x

2

1

1x

1cosh x

2

1; 1

1x

x

1tanh x2

1; 1

1

x

x

1coth x2

1; 1

1x

x

1sech x

2

1

; 0 11 xx x

cosech1x2

1; 0

1x

x x


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Generalised Form

( );( )

y f uu g x

dy dy dudx du dx

1sinh u

2

1

1

du

dxu

1

cosh u

2

1

; 11

du

udxu

1tanh u2

1; 1

1

duu

u dx

1coth u

21 ; 1

1du u

u dx

1sech u2

1; 0 1

1

duu

dxu u

cosech1u

2

1; 0

1

duu

dxu u


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Example 2.4: Find the derivatives of

(a) )31(sinh 1 xy

(b)

xy

1cosh 1

(c) xey x 1sech

(d) 1sinh (tan 3 )y x

(e)1 2tanh

( )1 sec

tf t

t

(f) uy

1

coth

(g)1cos 4 cosh 4y x x

(h) 3 1sinh 0y xy

chap 2 - differentiation

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