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CHAPTER 4

DIFFERENTIATION

4.1 Differentiation of hyperbolic functions

4.2 Differentiation of inverse trigonometric functions

4.3 Differentiation of inverse hyperbolic functions

*Recall: Methods of differentiation

-Chain rule -Product differentiation -Quotient differentiation -Implicit differentiation

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4.1 Differentiation of Hyperbolic Functions

Recall: Definition:

cosh2

x -xe e x

sinh2

x -xe e x

sinhtanhcosh

x -x

x -x

x e e x x e e

1 cosh coth tanh sinh

xxx x

1cosech = sinh

xx

1sechcosh

x x

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Derivatives of hyperbolic functions

Example 4.1: Find the derivatives of

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

Solution:

(a)

2sinh

xx eedxd

xdxd

xee xx cosh)(21

(b)

2cosh

xx eedxd

xdxd

xee xx sinh)(21

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(c)

xx

xx

ee

eedxd

xdxd

tanh

Using quotient diff:

2xx

xxxxxxxx

ee

eeeeeeee

2

2

2

2222

24

22

xxxx

xx

xxxx

eeee

ee

eeee

xx

22

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 x

dx

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 4.2:

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)

Find dydx

from the following expressions:

2(a) sinh 4 cosh

(b) tanh ( )x y x yy x y

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4.2 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)(sinx

xdxd

2. 2

1

1

1)(cosx

xdxd

3. 2

1

1

1)(tan

xx

dxd

4. 2

1

1

1)(cot

xx

dxd

5. 1

1)(sec

21

xxx

dxd

6. 1

1)(csc

21

xxx

dxd

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4.2.1 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 xdxdy

dxd

ydx

dydxdyy

cos1

1cos

Since 22

y , 0cos y , so

22 11

sin11

cos1

xyydxdy

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Example 4.3:

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 of 1 2tan (tan(3 1))y t .

4. Find the derivative dxdy if

(a) yxyx 21tan

(b) yxy 11 cos2

)(sin

11

Summary

If u is a differentiable function of x, then

1. dxdu

uu

dxd

21

1

1)(sin

2. dxdu

uu

dxd

21

1

1)(cos

3. dxdu

uu

dxd

21

1

1)(tan

4. dxdu

uu

dxd

21

1

1)(cot

5. dxdu

uuu

dxd

1

1)(sec

21

6. dxdu

uuu

dxd

1

1)(csc

21

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4.3 Derivatives of Inverse hyperbolic Functions Recall: Inverse Hyperbolic Functions

Function Domain Range

1sinhy x ),( ),(

1 coshy x ),1[ ),0[

1tanhy x (1, 1) ),(

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

y = sech1 x (0, 1] ),0[

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

Function Logarithmic form

1sinhy x 2ln 1x x

1 coshy x 2ln 1x x

1tanhy x 1 1ln ; 12 1

x xx

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4.3.1 Proof: ( 2

1

11)(sinh

xx

dxd

)

Recall: 1sinhy x sinhx y

To find its derivative we proceed implicitly:

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

)(sinh)( ydxdx

dxd

ydxdy

dxdyy

cosh1

cosh1

Since y , 0cosh y , so using the identity 2 2cosh sinh 1y y :

22 11

sin11

cosh1

xyydxdy

21

11)(sinh

xx

dxd

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Other ways to obtain the derivatives are:

(a) 1sinhy x sinhx y then

2

y ye ex

. Hence, find dydx

.

(b) 1sinhy x = 2ln 1x x .

Hence, find dydx

.

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Standard Derivatives

Function, y Derivatives, dydx

1sinh x 2

11x

1cosh x 2

1 ; 11

xx

1tanh x 2

1 ; 11

xx

1coth x 2

1 ; 11

xx

1sech x 2

1 ; 0 11

xx x

cosech1x 2

1 ; 01

xx x

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Generalised Form

( );( )

y f uu g x

dy dy dudx du dx

1sinh u

2

11

dudxu

1cosh u

2

1 ; 11

du udxu

1tanh u

2

1 ; 11

du uu dx

1coth u

2

1 ; 11

du uu dx

1sech u

2

1 ; 0 11

du udxu u

cosech1u

2

1 ; 01

du udxu u

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Example 4.4: Find the derivatives of

(a) )31(sinh 1 xy

(b)

xy 1cosh 1

(c) xey x 1sech

(d) 1sinh (tan3 )y x

(e) 1 2tanh( )

1 sectf tt

(f) uy 1coth

(g) 1cos 4 cosh 4y x x

(h) 3 1sinh 0y xy