Chapter 3
Derivatives
Section 1 Derivatives and Rates of Change
1
1.1 Review of tangent line and instatanious velocity
Definition
The tangent line is the limit position of the secant line.
y  f  x
y
f  x
 
tan   tan
tan   mPQ
f  x  f a

xa
Q
T
tan   mPT
f a
P

o
mPT

a
x
f  x  f a
 lim
x a
xa
f  a  h  f a 
 lim
h 0
h
x
2
Suppose s  f (t )
s t 
Q  a  h, f  a  h
change in position
average velocity=
time elapsed
P  a, f  a 
h
f  a  h  f  h
v  a   lim
h 0
h
a ah t
Definition The derivative of a function f at a number a, denoted by
f   a  is
f a  h  f a 
f  x  f a
f   a   lim
h 0
h
 lim
x a
xa
If this limit exists.
3
Example 1 Find the Derivative of the function f  x   x2  8x  9 at
the number a.
Example 2 Find an equation of the tangent line to the parabola
y  x 2  8x  9
at the point (3,-6).
4
Q  x2 , f  x2 
1.2 Rates of change
x  x2  x1
y  f  x2   f  x1 

P  x1, f  x1 

y f  x2   f  x1 

x
x2  x1
the slope of the secant line

the average velocity
y
x
x1
x2
is called the average rate of change of y with respect to x over the
interval  x1 , x2 
f  x2   f  x1  the slope of the tangent line
y
lim
 lim

x 0 x
x2  x1
x2  x1
the instantaneous velocity
5
The derivative f   a  is the instantaneous rate of change of y=f(x)
with respect to x when x=a
1.3 The interpretation of the instantaneous rate of change
Q


P
The derivative is large, the curve
is steep, the y-values change rapidly.
The derivative is small, the curve
is flat, the y-values change slowly.
6
Section 2 the Derivatives As a Function
2.1 The derivative of a function
f a  h  f a 
f   a   lim
h 0
h
is called the derivative at a fixed number a.
f  x  h  f  x
f   x   lim
is called the derivative of f .
h 0
h
Given any number x for which this limit exits, we assign x to
the number f   x 
7
Example 1 The graph of a function f is given in the following graph,
use it to sketch the graph of the derivative f   x 
B
P 
A
C
8
2.2 Other notation
f   x   lim
h 0
f   x   y 
f  x  h  f  x
h
dy df
d


f  Df  x   Dx f  x 
dx dx dx
d
are called differentiation operators, which is the process of
dx
calculating a derivative.
D and
Definition A function f is differentiable at a if f   a  exisits.
It is differentiable on an open interval (a,b) if it is
differentiable at every number in the interval.
9
Example 2
(a) If f  x   x3  x , find f   x  and f  1 .
(b) Illustrate by comparing the graph of f and f   x 
Example 3 If f  x   x ,find the derivative of f . State the domain
of f   x 
Example 4
Find f   x  if f  x  
1 x
2 x
10
2.3 the relationship between continuous and differentiable
Theorem If f is differentiable at a, then f is continuous at a.
Conclusion (1)The inverse proposition is not ture, that is if f is
continuous at a, the function is not necessary
differentiable at a.
(2)But the inverse negative proposition is true, that is if
f is not continuous at a, then f is not differentiable at
a definitely.
Example 5 Where is the function f  x   x differentiable?
11
2.4 How can a function fail to be differentiable?

a
A corner
a
A discontinuity
a
A vertical tangent
Three ways for f not to be differentiable at a
12
2.5 Higher derivatives
Definition If f is a differentiable function, and its derivative f  have
derivative of its own, denoted by  f   f , then this new function
f  is called the second derivative of f .
v t   S  t 
d  dy  d 2 y
f      2
dx  dx  dx
a t   v t   S t 
d3y
f   3
dx
f  n
dny
 n
dx
3
Example6 If f  x   x  x ,find f 
13
Section 3 Differentiation Formulas
Derivative of a constant function
d
c  0
dx
Derivative of a power function
d
 x  1
dx
d n
x   nx n 1 , here n is a positive integral.

dx
14
The constant multiple rule
If c is a constant and f is a derivative function, then
d
d
f  x
cf  x    c
dx
dx
Differentiation rules
Suppose f and g are differentiable
functions , then their sum, difference, product and quotient are also
differentiable functions, and we have
(1) [ f ( x)  g ( x)]  f ( x)  g ( x);
(2) [ f ( x)  g ( x)]  f ( x) g ( x)  f ( x) g ( x);
f ( x)
f ( x) g ( x)  g ( x) f ( x)
(3) [
] 
( g ( x)  0).
2
g ( x)
g ( x)
15
d 8
x  12 x5  4 x 4  10 x3  6 x  5 
Example 7 Find

dx
4
2
Example 8 Find the points on the curve y  x  6x  4
where the tangent line is horizontal.
Example 9 Find F   x  if F  x    6 x3  7 x 4 
Example 10
If h  x  =xg  x  and it is known that g  3 =5 and g   3 =2,
find h  3 .
x2  x  2
Example 11 Let y  x3  6 , find y.
16
Section 4 Derivative of trigonometric functions
Preparation
(a)some trigonomitric equalities
 b  lim
 0
sin 

1
d
 sin x   cos x
dx
d
 cos x    sin x
dx
d
 tan x   sec2 x
dx
d
 cot x    csc2 x
dx
d
 sec x   sec x tan x
dx
d
 csc x    csc x cot x
dx
17
Example 12 Find the 27th derivative of cos x
Example 13 Find lim
x0
sin 7 x
.
4x
x cot x.
Example 14 Calculate lim
x0
18
Section 5
The chain rule
The Chain Rule
If g is differentiable at x and f is differentiable at
g(x), then the composite function F(x)=f(g(x))
is differentiable at x and F  is given by the
product
F   x   f   g  x  g  x 
That means, if y=f(u) and u=g(x) are both differentiable function,
then
dy dy du
dx


du dx
When we use this formula we should bear in mind dy/dx refers to the
derivative of y with respect to x, (i.e. y should be regarded as a function
of x). dy/du refers to the derivative of y with respect to u, (i.e. y should
be regarded as a function of u).
19
Example 15 Suppose y=sin2x, find dy/dx.
Example 16 Differentiate
a 
y  sin  x 2  and
 b  y  sin 2 x.
Example 17 Differentiate y   x  1 .
3
Example 18 Differentiate
Example 19 Differentiate
100
y   2 x  1  x  x  1 .
5
3
4
y  sin  cos  tan x .
20
The power rule (general version)
If n is any number, then
d n
x   nx n 1.

dx
Example 20 Differentiate the following functions.
a  f  x  x

b y  3
1
x2
c F  x  
x2  1
d  y 
sec x3
Example 21 Find equation of the tangent line and normal line to the
curve y  x / 1  x 2  at the point (1,1/2).
21
Section 6
Implicit Function
So far ,we have met function can be discribed byexpressing one
variable explicitly in terms of another variable.
y  x2  1
y  sin x
y  f  x
However, some functions are defined implicitly by a relation
between x and y such as
x2  y2  25
x3  y3  6xy
1
 2
Folium of Decartes.
How to find the derivative of y without solving an equation y
in terms of x.
22
Implicit Differentiation
Step 1. Differentiate both side of the equation with respect to x.
Step 2. Solve the resulting equation for y.
Example 1. (a) If x 2  y 2  25, find
dy
.
dx
(b) Find an equation of the tangent to the circle
x2  y 2  25 at the point (3, 4).
23
Example 2. (a) If x3  y 3  6 xy, find
dy
.
dx
(b) Find the equation of the tangent to the folium of
Decartes at the point (3,3).
(c) At what point in the first quadrant is the tangent line
horizontal?
2

y
if
sin
x

y

y
cos x.


Example 3. Find
4
4
Example 4. Find y if x  y  16.
24
Section 9 Linear Approaximations
and Differentials
y  f  x
y
f  x
If f is differentiable at
a, then when we zoom
in toward the point (a,f(a))
The graph straightens out
and appears more and more
like a line.
Q
T
f a
P




o
a
x
x
This observation is the basis
for a method of finding
approximate values of
functions
25
That means we can use the tangent line at (a,f(a)) as an approximation
to the curve y=f(x) when x is near a.
y  L  x   f  a   f   a  x  a 
y
y  f  x
linearization of f at a
T
P (a,f(a))
f a
f  x   f  a   f   a  x  a 
y  L  x


o
a
x
linear approximation of f at a
or tangent line approximation
26
Example 5 Find the linearization of the function f  x   x  3 at a=1,
and use it to approximate the numbers 3.98 and 4.05.
Are these approximations overestimates or underestimates?
3.98  1.995 and
4.05  2.0125.
x
From L(x)
Actual value
3 .9
0 .9
1.975
1.97484176
3.98
0.98
1.99499373
4
1
1.995
2
4.05
2.0125
4 .1
1.05
1 .1
5
2
2.025
2.25
2.01246117
2.02484567
6
3
2 .5
2.00000000
2.23606797
2.44948974
27
Differential
the ideas behind the linear approximation is differential.
R
y  f  x  x   f  x 
Q
P
y
dy
dy  f   x  x  f   x  dx
dx  x S
dy it is called the differential
x
x  x
f  x  x   f  x   dy
when x is very small, y  dy.
f  x  x   f  x   dy
We can use the value of the function at x plus dy to approximate
the value of the function at x  x.
Because dy is a linear function, it is easier to calculate than y .
28
Example 6
Compare the values of
y and dy if
y  f  x   x3  x2  2x  1 and x changes (a) from 2 to 2.05
(b) from 2 to 2.01
f  x   f  a   f   a  x  a 
x  a  dx
x  a  dx
f  a  dx   f  a   f   a  dx
f  a  dx   f  a   dy
Example 7 Using the linear approximation to estimate tan44。
29
Example 5 The radius of sphere was measured and found to be 21cm,
with the possible error in measurement of at most 0.05cm.
Find the approximation to the volume and the relative error.
30