Post on 27-Dec-2015
transcript
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