derivatives of polynomials derivative of a constant function we have proved the power rule we can...
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Derivatives of polynomials Derivative of a constant function
We have proved the power rule
We can prove
1( )n ndx nx
dx
( ) 0d
cdx
2
1 1( )
d
dx 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:
)())(( xfdx
dcxcf
dx
d
)()()]()([ xgdx
dxf
dx
dxgxf
dx
d
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 hx x
h h
a a af x a a f
h h
( ) xf x a
( )x xde e
dx
0
1lim 1
h
h
e
h
Product rule for derivative
The product rule:
g is differentiable, thus continuous, therefore,
)()()()()]()([ xfdx
dxgxg
dx
dxfxgxf
dx
d
( ) ( ) ( ) ( ) ( )
[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]
( ) ( ) ,
( )( ) ( ) .
fg f x x g x x f x g x
f 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 g
fg f gg x x f x
x 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 x
x 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:
)()()()()()()()()(
))()()(())()()(())()()((
xhxgxfxhxgxfxhxgxf
xhxgxfxhxgxfxhxgxf
Example Find if
Sol.
)(xf 2( ) .xf x x e
.)2(2)()()( 2222 xxxxx exxexxeexexxf
Quotient rule for derivativeThe quotient rule: .
)(
)()()()(
)(
)(2 xg
xgxfxfxg
xg
xf
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ( ) ) ( )( ( ) )
( / ) ( ) ( ).
( )( ( ) ) ( )( ( ) )
f f x x f x f x f f x
g 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 g
g x g x g g x g x g
f g g x f f x g
x 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
x
n
xx
)(
)(
)(
12 xf
xf
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 xy
x
3 2 2 3
3 2
( 6)( 2) ( 2)( 6)
( 6)
x x x x x xy
x
3 2 2
3 2
( 6)(2 1) ( 2)(3 )
( 6)
x x x x x
x
4 3 2
3 2
2 6 12 6
( 6)
x x x x
x
Table of differentiation formulas
( ) 0d
cdx
1( )n ndx nx
dx ( )x xd
e edx
( )cf cf ( )f g f g
( )fg fg gf
2
f gf fg
g g
An important limit Prove that
Sol. 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 x
xsin
cos 1, ( / 2,0) (0, / 2)x
x 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 lim
2 ( / 2)
sincos lim cos
h h
h h h
t
f x h f x x h xf x
h hx h h
x h h
h h
tx 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 lim
2 ( / 2)
sinsin lim sin
h h
h h h
t
f x h f x x h xf x
h hx h h
x h h
h h
tx 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 t
h t
Example
Ex. Evaluate the limit
Sol. Using the formula
and putting u=(x-a)/2, we derive
.sinsin
limax
axax
.cos2
sin2lim
2coslim
2sin
2cos2
limsinsin
lim
0a
u
uaxax
axax
ax
ax
uax
axax
2sin
2cos2sinsin
axaxax
Example
Ex. Find the limit
Sol. Using the trigonometry identity
and putting u=x/2, we obtain
.cos1
lim20 x
xx
2 2
2 2 20 0 0
1 cos 2sin ( / 2) sinlim lim lim
2x x u
x x u
x x u
2sin2cos1 2 x
x
2 2
0 0
1 sin 1 sin 1lim lim .
2 2 2x x
u u
u u
Example Ex. Find the limits: (a) (b)
Sol. (a) Letting then and
(b) Letting then
,arcsin
lim0 x
xx
.
2
coslim
2 x
x
x
.1sin
limarcsin
lim00
u
u
x
xux
02 2
sin( )cos sin2lim lim lim 1.
2 2
ux x
xx u
ux x
arcsin ,u x sin ,x u
/ 2 ,u x