Derivatives of polynomials Derivative of a constant function

We have proved the power rule

We can prove

1( )n nd x nxdx

( ) 0d cdx

2

1 1( )ddx x x

20 0

1 11 1 1( ) lim lim

( )h h

x h xx h x x h x

Rules for derivative The constant multiple rule:

The sum/difference rule:

)())(( xfdxdcxcf

dxd

)()()]()([ xgdxdxf

dxdxgxf

dxd

Exponential functions Derivative of

The rate of change of any exponential function is proportional to the function itself.

e is the number such that Derivative of the natural exponential function

0 0

1( ) lim lim (0)x h x h

x x

h h

a a af x a a fh h

( ) xf x a

( )x xd e edx

0

1lim 1h

h

eh

Product rule for derivativeThe product rule:

g is differentiable, thus continuous, therefore,

)()()()()]()([ xfdxdxgxg

dxdxfxgxf

dxd

( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]

( ) ( ) ,( ) ( ) ( ) .

fg f x x g x x f x g xf x x g x x f x g x x f x g x x f x g x

g x x f f x gfg f gg x x f xx x x

0 0 0 0

( )lim lim ( ) lim ( ) lim ( ) ( ) ( ) ( ).x x x x

fg f gg x x f x g x f x f x g xx x x

Remark on product rule In words, the product rule says that the derivative of a

product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Derivative of a product of three functions:

)()()()()()()()()())()()(())()()(())()()((

xhxgxfxhxgxfxhxgxfxhxgxfxhxgxfxhxgxf

Example Find if

Sol.

)(xf 2( ) .xf x x e

.)2(2)()()( 2222 xxxxx exxexxeexexxf

Quotient rule for derivativeThe quotient rule: .

)()()()()(

)()(

2 xgxgxfxfxg

xgxf

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ( ) ) ( )( ( ) )

( / ) ( ) ( ) .( )( ( ) ) ( )( ( ) )

f f x x f x f x f f xg g x x g x g x g g x

f x g x g x f f x g x f x g g x f f x gg x g x g g x g x g

f g g x f f x gx g x g x g x g x g x g x

Example Using the quotient rule, we have:

which means

is also true for any negative integer k.

1)(1 )(1)(

n

nnn xn

xn

xx

)()(

)(1

2 xfxf

xf

1)( kk kxx

Homework 4 Section 2.7: 8, 10

Section 2.8: 16, 17, 22, 24, 36

Section 2.9: 28, 30, 46, 47

Page 181: 13

Example We can compute the derivative of any rational functions.

Ex. Differentiate

Sol.

2

3

2 .6

x xyx

3 2 2 3

3 2

( 6)( 2) ( 2)( 6)( 6)

x x x x x xyx

3 2 2

3 2

( 6)(2 1) ( 2)(3 )( 6)

x x x x xx

4 3 2

3 2

2 6 12 6( 6)

x x x xx

Table of differentiation formulas

( ) 0d cdx

1( )n nd x nxdx

( )x xd e edx

( )cf cf ( )f g f g

( )fg fg gf

2

f gf fgg g

An important limit Prove thatSol. It is clear that when thus Since and are even functions,we have Now the squeeze theorem together with

gives the desired result.

(0, ), sin tan2

x x x x

cos x sin xx

sincos 1, ( / 2,0) (0, / 2)xx xx

Derivative of sine functionFind the derivative of Sol. By definition,

( ) sin .f x x

0 0

0 0 0

0

( ) ( ) sin( ) sin( ) lim lim

22cos sin 2 sin( / 2)2 2lim lim cos lim2 ( / 2)

sincos lim cos

h h

h h h

t

f x h f x x h xf xh h

x h hx h h

h htx x

t

Derivative of cosine functionEx. Find the derivative of Sol. By definition,

.cos)( xxf

0 0

0 0 0

0

( ) ( ) cos( ) cos( ) lim lim

22sin sin 2 sin( / 2)2 2lim limsin lim2 ( / 2)

sinsin lim sin

h h

h h h

t

f x h f x x h xf xh h

x h hx h h

h htx x

t

Derivatives of trigonometric functions

Using the quotient rule, we have:

(sec ) sec tan , (csc ) csc cotx x x x x x

2(tan ) sec ,x x 2(cot ) cscx x

Change of variable The technique we use in

is useful in finding a limit.

The general rule for change of variable is:

).(lim))((lim )()( ufxgflu

axlxg

ax

0 0

sin( / 2) sinlim lim 1( / 2)h t

h th t

Example Ex. Evaluate the limit

Sol. Using the formula and putting u=(x-a)/2, we derive

.sinsinlimax

axax

.cos2sin2lim

2coslim

2sin

2cos2

limsinsinlim

0a

uuax

ax

axax

axax

uax

axax

2sin

2cos2sinsin axaxax

Example Ex. Find the limit

Sol. Using the trigonometry identity and putting u=x/2, we obtain

.cos1lim 20 xx

x

2 2

2 2 20 0 0

1 cos 2sin ( / 2) sinlim lim lim2x x u

x x ux x u

2sin2cos1 2 xx

2 2

0 0

1 sin 1 sin 1lim lim .2 2 2x x

u uu u

Example Ex. Find the limits: (a) (b)

Sol. (a) Letting then and

(b) Letting then

,arcsinlim0 x

xx

.

2

coslim2 x

xx

.1sin

limarcsinlim00

u

ux

xux

02 2

sin( )cos sin2lim lim lim 1.

2 2ux x

xx uux x

arcsin ,u x sin ,x u

/ 2 ,u x