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CHAPTER 4

DIFFERENTIATION

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INTRODUCTION • Differentiation

– Process of finding the derivative of a function.

• Notation

yDydxdy

xfxfdxd

x,,

,

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DERIVATIVE OF A POWER FUNCTION • If n is an integer, then:

1 nn nxxdxd

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DERIVATIVE OF A CONSTANT If f is differentiable at function x and c is any real number, then c is differentiable:

xfdxdcxcf

dxd

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Example 1Differentiate the following function:

10

2

3

3

4

2

xxfc

xxfb

xxfa

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DERIVATIVE OF SUM AND DIFFERENCE RULES

If f and g are differentiable at function x, then the function f+g and f-g are differentiable:

xgdxdxf

dxdxgxf

dxd

xgdxdxf

dxdxgxf

dxd

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Example 2Differentiate the following function:

20

3

2

5

23

3

xxfc

xxxfb

xxxfa

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Derivative of Trigonometric Functions

xfy )(xfdxdy

xsin

xcos

kxsin

kxcos

xcos

xsin

kxk cos

kxk sin

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DERIVATIVE OF EXPONENTIAL & LOGARITHMIC FUNCTIONS

xfy )(xfdxdy

xe

xln x1

xeaxe axae

axln x

axdxd

ax11

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PRODUCT RULEIf u and v are differentiable at function x, then so the product u.v, thus

udxdvv

dxduuv

dxd

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Example 3:Differentiate the following function:

xeyd

xeycxxyb

xxxya

x

x

5ln2

4sin32

3

3

2

42

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QUOTIENT RULEIf u and v are differentiable at function x, then is also differentiable

2v

vdxduu

dxdv

vu

dxd

vu

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Example 4Differentiate the following function:

x

eycx

xybxx

xya

x

3sin

2cos

21

4

33

2

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Example 5Differentiate and SIMPLIFY the following function:

xxyd

xeyc

xxyb

xeya

x

x

3cos12sin

sin4

2sin2

cos

3

3

2

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Example 6Differentiate the following function:

xeydx

eyc

exyb

xxya

x

x

x

3ln3sin

2

ln

3

2

33

2

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COMPOSITE FUNCTIONThe Chain Rule

– If g is differentiable at point x and f is differentiable at the point g(x), then is differentiable at x.

– Let and , then xgfy

dxdu

dudy

dxdy

gf xgu

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Example 7Differentiate the following function:

xyb

xya

3sin

13ln

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“Outside-Inside” Rule– Alternative method for Chain Rule:– If ,then xgfy

xgxgfdxdy .

COMPOSITE FUNCTION

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Example 8Differentiate the following function:

x

x

exyc

eyb

xya

22

2sin

ln

3sinln

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• These equation define an implicit relation between variables x and y.

• When we cannot put an equation F(x,y)=0 in the form y = f(x), use implicit differentiation to find

IMPLICIT DIFFERENTIATION

09,025 3322 xyyxyx

dxdy

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• Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x

• Collect the terms with on one side of the equation

• Solve for

IMPLICIT DIFFERENTIATION

dxdy

dxdy