Lesson 8: Derivatives of Polynomials and Exponential functions
Derivatives of polynomials
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Transcript of Derivatives of polynomials
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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
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Rules for derivative The constant multiple rule:
The sum/difference rule:
)())(( xfdxdcxcf
dxd
)()()]()([ xgdxdxf
dxdxgxf
dxd
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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
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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
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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
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Example Find if
Sol.
)(xf 2( ) .xf x x e
.)2(2)()()( 2222 xxxxx exxexxeexexxf
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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