Derivative Part 2. Definition The derivative of a function f is another function f ’ (read “f...
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Transcript of Derivative Part 2. Definition The derivative of a function f is another function f ’ (read “f...
![Page 1: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/1.jpg)
Derivative
Part 2
![Page 2: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/2.jpg)
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
![Page 3: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/3.jpg)
Soo
if x1 belong to domain then
x
xfxxf) f'(x
x
)(lim 11
01
![Page 4: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/4.jpg)
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
![Page 5: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/5.jpg)
Differentiability Implies Continiuty
Ex. Check if continue at x=0 and differentiable at x=0?
xxf )(
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The Constant RuleThe Constant Rule
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The Power RuleThe Power Rule
![Page 8: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/8.jpg)
The Constant Multiple RuleThe Constant Multiple Rule
![Page 9: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/9.jpg)
The Sum and Difference RulesThe Sum and Difference Rules
![Page 10: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/10.jpg)
Derivatives of Sine and Cosine FunctionsDerivatives of Sine and Cosine Functions
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The Product RuleThe Product Rule
![Page 12: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/12.jpg)
The Quotient RuleThe Quotient Rule
![Page 13: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/13.jpg)
Derivatives of Trigonometric Derivatives of Trigonometric FunctionFunction
![Page 14: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/14.jpg)
Leibniz Notation for Derivatives
Ultimately, this notation is a better and more effective notation for working with derivatives.
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Teorema
If and differentiable function then )(xu )(xv
)(')('
)()(
')()()()( .1
xvxudx
xdv
dx
xdu
xvxuxvxudx
d
![Page 16: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/16.jpg)
)(')(
')()(constan,.3
)()(')(')(
)()(
)()(
')().()().( .2
xkudx
xduk
xkuxkudx
dk
xvxuxvxudx
xduxv
dx
xdvxu
xvxuxvxudx
d
![Page 17: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/17.jpg)
2
2
'
)(
)(')()()('
)(
)()(
)()(
)(
)(
)(
)(.4
xv
xvxuxvxu
xvdx
xdvxu
dxxdu
xv
xv
xu
xv
xu
dx
d
![Page 18: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/18.jpg)
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
![Page 19: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/19.jpg)
The Chain RuleThe Chain Rule
![Page 20: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/20.jpg)
The General Power RuleThe General Power Rule
![Page 21: 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.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649e495503460f94b3c9a9/html5/thumbnails/21.jpg)
Summary of Differentiation RulesSummary of Differentiation Rules
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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
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Exercise 2
Check if the function
differentiable at 0 ??
0,0
0,1
sin)(x
xxxg
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Ex3
Check if the function
Differentiable at x=0
xxxg sin)(
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Ex 4
Find the derivative from the function :
xxf )(
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Ex 5
Calculate d/dx(x) then show the function y= x satisfied yy’=x, x0
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Ex 6
Find the derivative from the invers function
2,4)( 2 xxxxf