Ch.2 Limits and derivatives
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Transcript Ch.2 Limits and derivatives
Derivatives of polynomials
d
Derivative of a constant function (c) 0
dx
d n
n 1
(
x
)
nx
We have proved the power rule
dx
We can prove
d 1
1
( ) 2
dx x
x
1
1
1
1
1
x
h
x
( ) lim
lim
2
h 0
h 0 x ( x h)
x
h
x
Rules for derivative
The constant multiple rule:
d
d
(cf ( x)) c
f ( x)
dx
dx
The sum/difference rule:
d
d
d
[ f ( x) g ( x)]
f ( x) g ( x)
dx
dx
dx
Exponential functions
Derivative of
xh
f ( x) lim
h 0
a
f ( x) a x
ax
ah 1
x
a lim
a x f (0)
h 0
h
h
The rate of change of any exponential function is
proportional to the function itself.
eh 1
1
e is the number such that lim
h 0
h
Derivative of the natural exponential function
d x
(e ) e x
dx
Product rule for derivative
d
d
d
The product rule: [ f ( x) g ( x)] f ( x) g ( x) g ( x) f ( x)
dx
dx
dx
( 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
g
g ( x x) f ( x) .
x
x
x
g is differentiable, thus continuous, therefore,
( fg )
f
g
lim
lim g ( x x) lim f ( x) lim g ( x) f ( x) f ( x) g ( x).
x 0 x
x 0
x 0 x
x 0 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:
( f ( x) g ( x)h( x)) f ( x)(g ( x)h( x)) f ( x)(g ( x)h( x))
f ( x) g ( x)h( x) f ( x) g ( x)h( x) f ( x) g ( x)h( x)
Example
Find f (x ) if f ( x) x2e x .
Sol. f ( x) ( x2 )e x x2 (e x ) 2xex x2e x ( x 2 2x)e x .
Quotient rule for derivative
The quotient rule: f ( x) g ( x) f ( x) f ( x) g ( x) .
2
g ( x)
g
( x)
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:
1
f ( x)
2
f ( x)
f ( x)
n
1
( x n ) n n 1 ( n) x ( n ) 1 which means ( x k ) kxk 1
x
x
is also true for any negative integer k.
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.
x2 x 2
.
Ex. Differentiate y
3
x 6
3
2
2
3
(
x
6)(
x
x
2)
(
x
x
2)(
x
6)
Sol. y
( x3 6)2
( x3 6)(2 x 1) ( x 2 x 2)(3x 2 )
( x3 6)2
x 4 2 x3 6 x 2 12 x 6
( x3 6)2
Table of differentiation formulas
d
(c ) 0
dx
(cf ) cf
d n
( x ) nx n 1
dx
d x
(e ) e x
dx
( f g ) f g
( fg ) fg gf
f gf fg
2
g
g
An important limit
Prove that
Sol. It is clear that when x (0, ), sin x x tan x
2
thus
Since cos x and sin x are even functions,
x
we have
sin x
cos x
1, x ( / 2, 0) (0, / 2)
x
Now the squeeze theorem together with
gives the desired result.
Derivative of sine function
Find the derivative of f ( x) sin x.
Sol. By definition,
f ( x h) f ( x )
sin( x h) sin x
f ( x) lim
lim
h 0
h 0
h
h
2x h
h
2 cos
sin
2x h
sin(h / 2)
2
2
lim
lim cos
lim
h 0
h 0
h 0 ( h / 2)
h
2
sin t
cos x lim
cos x
t 0
t
Derivative of cosine function
Ex. Find the derivative of f ( x) cos x.
Sol. By definition,
f ( x h) f ( x )
cos( x h) cos x
lim
h 0
h 0
h
h
2x h
h
2sin
sin
2x h
sin(h / 2)
2
2
lim
limsin
lim
h 0
h
0
h 0 ( h / 2)
h
2
sin t
sin x lim
sin x
t 0
t
f ( x) lim
Derivatives of trigonometric functions
Using the quotient rule, we have:
(sec x) sec x tan x, (csc x) csc x cot x
(tan x) sec2 x,
(cot x) csc2 x
Change of variable
The technique we use in
sin(h / 2)
sin t
lim
lim
1
h 0 ( h / 2)
t 0
t
is useful in finding a limit.
The general rule for change of variable is:
g ( x )l ( x a )
lim f ( g ( x)) lim f (u ).
x a
u l
Example
Ex. Evaluate the limit
lim
xa
sin x sin a
.
xa
xa xa
sin
Sol. Using the formula sin x sin a 2 cos
2
2
and putting u=(x-a)/2, we derive
xa
xa
2 cos
sin
sin x sin a
2
2
lim
lim
x a
x a
xa
xa
xa
2 sin u
lim cos
lim
cos a.
x a
u 0
2
2u
Example
1 cos x
.
Ex. Find the limit lim
2
x 0
x
Sol. Using the trigonometry identity 1 cos x 2 sin 2
and putting u=x/2, we obtain
1 cos x
2 sin 2 ( x / 2)
sin 2 u
lim
lim
lim
2
2
x 0
x
0
u 0 2u 2
x
x
2
2
1
1
sin u
1
sin u
lim
lim
.
x
0
x
0
2
2
u
2
u
x
2
Example
cos x
arcsin x
lim
.
Ex. Find the limits: (a) lim
, (b)
x0
x
x
x
2
2
Sol. (a) Letting u arcsin x, then x sin u, and
arcsin x
u
lim
lim
1.
x 0
u
0
x
sin u
(b) Letting u / 2 x,
lim
x
2
cos x
2
x
lim
x
2
sin(
2
2
then
x)
x
sin u
lim
1.
u 0
u