Derivative

Part 2

Definition

The derivative of a function f is another function f ’ (read “f prime”) whose value at any number x is :

Provided that this limit exists and is not or -

x

xfxxf'(x) f

x

)(lim 00

0

If this limit does exist f differentiable at cOther way if f differentiable at x1 then f ‘(x1) existIf a function differentiable at every riil number in their domain then f called differentiable function

Soo

if x1 belong to domain then

x

xfxxf) f'(x

x

)(lim 11

01

Add

Note : If we take

then

x

xfxxf) f'(x

x

)(lim 11

01

11 ~0 xxxxxx

1

1

11

)()(lim)('

xx

xfxfxf

xx

3

2)( if )(' find

Example

xxgcg

Differentiability Implies Continiuty

Ex. Check if continue at x=0 and differentiable at x=0?

xxf )(

The Constant RuleThe Constant Rule

The Power RuleThe Power Rule

The Constant Multiple RuleThe Constant Multiple Rule

The Sum and Difference RulesThe Sum and Difference Rules

Derivatives of Sine and Cosine FunctionsDerivatives of Sine and Cosine Functions

The Product RuleThe Product Rule

The Quotient RuleThe Quotient Rule

Derivatives of Trigonometric Derivatives of Trigonometric FunctionFunction

Leibniz Notation for Derivatives

Ultimately, this notation is a better and more effective notation for working with derivatives.

Teorema

If and differentiable function then )(xu )(xv

)(')('

)()(

')()()()( .1

xvxudx

xdv

dx

xdu

xvxuxvxudx

d

)(')(

')()(constan,.3

)()(')(')(

)()(

)()(

')().()().( .2

xkudx

xduk

xkuxkudx

dk

xvxuxvxudx

xduxv

dx

xdvxu

xvxuxvxudx

d

2

2

'

)(

)(')()()('

)(

)()(

)()(

)(

)(

)(

)(.4

xv

xvxuxvxu

xvdx

xdvxu

dxxdu

xv

xv

xu

xv

xu

dx

d

dydxdx

dyxfxfy

dx

du

du

dy

dx

dyxguufy

1)( inversan have then )( if.6

)(),( if.5

1

1

1

)('0,,)( jika

)(',)( jika .7

nn

nn

nxxfxZnxxf

nxxfZnxxf

The Chain RuleThe Chain Rule

The General Power RuleThe General Power Rule

Summary of Differentiation RulesSummary of Differentiation Rules

Exercise 1

Suppose f with

Find a and b such as f continue at x=0 but f’(0) does’nt exist

0,

0,)( 2 xx

xbaxxf

Exercise 2

Check if the function

differentiable at 0 ??

0,0

0,1

sin)(x

xxxg

Ex3

Check if the function

Differentiable at x=0

xxxg sin)(

Ex 4

Find the derivative from the function :

xxf )(

Ex 5

Calculate d/dx(x) then show the function y= x satisfied yy’=x, x0

Ex 6

Find the derivative from the invers function

2,4)( 2 xxxxf