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3DIFFERENTIATION RULESDIFFERENTIATION RULES
We have:
Seen how to interpret derivatives as slopes and rates of change
Seen how to estimate derivatives of functions given by tables of values
Learned how to graph derivatives of functions that are defined graphically
Used the definition of a derivative to calculate the derivatives of functions defined by formulas
DIFFERENTIATION RULES
These differentiation rules enable
us to calculate with relative ease
the derivatives of: Polynomials
Rational functions
Algebraic functions
Exponential and logarithmic functions
Trigonometric and inverse trigonometric functions
DIFFERENTIATION RULES
3.1Derivatives of Polynomials
and Exponential Functions
In this section, we will learn:
How to differentiate constant functions, power functions,
polynomials, and exponential functions.
DIFFERENTIATION RULES
Let’s start with the simplest of
all functions—the constant function
f(x) = c.
CONSTANT FUNCTION
The graph of this function is the
horizontal line y = c, which has slope 0.
So, we must have f’(x) = 0.
CONSTANT FUNCTION
A formal proof—from the definition of
a derivative—is also easy.
0
0 0
( )'( ) lim
lim lim0 0
h
h h
f x h f xf x
hc c
h
CONSTANT FUNCTION
We next look at the functions
f(x) = xn, where n is a positive
integer.
POWER FUNCTIONS
If n = 1, the graph of f(x) = x is the line
y = x, which has slope 1.
So,
You can also verify Equation 1 from the definition of a derivative.
( ) 1dx
dx
POWER FUNCTIONS Equation 1
We have already investigated the cases
n = 2 and n = 3.
In fact, in Section 2.8, we found that:
2 3 2( ) 2 ( ) 3d dx x x x
dx dx
POWER FUNCTIONS Equation 2
For n = 4 we find the derivative of f(x) = x4
as follows:
4 4
0 0
4 3 2 2 3 4 4
0
3 2 2 3 4
0
3 2 2 3 3
0
( ) ( ) ( )'( ) lim lim
4 6 4lim
4 6 4lim
lim 4 6 4 4
h h
h
h
h
f x h f x x h xf x
h h
x x h x h xh h x
h
x h x h xh h
h
x x h xh h x
POWER FUNCTIONS
If n is a positive integer,
then1( )n nd
x nxdx
POWER RULE
In finding the derivative of x4, we had to
expand (x + h)4.
Here, we need to expand (x + h)n .
To do so, we use the Binomial Theorem—as follows.
0 0
( ) ( ) ( )'( ) lim lim
n n
h h
f x h f x x h xf x
h h
Proof 2POWER RULE
This is because every term except the first has h as a factor and therefore approaches 0.
1 2 2 1
0
1 2 2 1
0
1 2 2 1 1
0
( 1)2
'( ) lim
( 1)2lim
( 1)lim
2
n n n n n n
h
n n n n
h
n n n n n
h
n nx nx h x h nxh h x
f xh
n nnx h x h nxh h
hn n
nx x h nxh h nx
POWER RULE Proof 2
a.If f(x) = x6, then f’(x) = 6x5
b.If y = x1000, then y’ = 1000x999
c.If y = t4, then
d. = 3r23( )
dr
dr
Example 1POWER RULE
34dy
tdt
What about power functions with
negative integer exponents?
In Exercise 61, we ask you to verify from the definition of a derivative that:
We can rewrite this equation as:
2
1 1d
dx x x
1 21dx x
dx
NEGATIVE INTEGERS
What if the exponent is a fraction?
In Example 3 in Section 2.8, we found that:
This can be written as:
1
2
dx
dx x
1 2 1 212
dx x
dx
FRACTIONS
If n is any real number,
then1( )n nd
x nxdx
POWER RULE—GENERAL VERSION
Differentiate:
a.
b.
In each case, we rewrite the function as a power of x.
2
1( )f x
x
3 2y x
Example 2POWER RULE
Since f(x) = x-2, we use the Power Rule
with n = -2:
Example 2 a
2
2 1
33
'( ) ( )
2
22
df x x
dx
x
xx
POWER RULE
3 2
2 3
2 3 1 1 323
( )
( )
2
3
dy dx
dx dxdx
dx
x x
POWER RULE Example 2 b
It also enables us to find normal
lines.
The normal line to a curve C at a point P is the line through P that is perpendicular to the tangent line at P.
In the study of optics, one needs to consider the angle between a light ray and the normal line to a lens.
NORMAL LINES
Find equations of the tangent line and normal
line to the curve at the point (1, 1).
Illustrate by graphing the curve and these
lines.
Example 3TANGENT AND NORMAL LINES
y x x
The derivative of is:
So, the slope of the tangent line at (1, 1) is:
Thus, an equation of the tangent line is:
or
321 2( )f x x x xx x
3 2 1 1/ 23 3 32 2 2'( )f x x x x
32'(1)f
321 ( 1)y x 3 1
2 2y x
Example 3TANGENT LINE
The normal line is perpendicular to
the tangent line.
So, its slope is the negative reciprocal of , that is .
Thus, an equation of the normal line is:
or231 ( 1)y x 52
3 3y x
Example 3NORMAL LINE
322
3
We graph the curve and its tangent line
and normal line here.
Example 3TANGENT AND NORMAL LINES
If c is a constant and f is a differentiable
function, then
( ) ( )d dcf x c f x
dx dx
CONSTANT MULTIPLE RULE
Let g(x) = cf(x).
Then,0
0
0
0
( ) ( )'( ) lim
( ) ( )lim
( ) ( )lim
( ) ( )lim (Law 3 of limits)
'( )
h
h
h
h
g x h g xg x
hcf x h cf x
hf x h f x
ch
f x h f xc
hcf x
ProofCONSTANT MULTIPLE RULE
4 4
3 3
.a. (3 ) 3 ( )
3(4 ) 12
b. ( ) ( 1)
1 ( ) 1(1) 1
d dx x
dx dx
x x
d dx x
dx dxdx
dx
NEW DERIVATIVES FROM OLD Example 4
If f and g are both differentiable,
then
( ) ( ) ( ) ( )d d df x g x f x g x
dx dx dx
SUM RULE
Let F(x) = f(x) + g(x). Then,
0
0
0
0 0
( ) ( )'( ) lim
( ) ( ) ( ) ( )lim
( ) ( ) ( ) ( )lim
( ) ( ) ( ) ( )lim lim (Law 1)
'( ) '( )
h
h
h
h h
F x h F xF x
hf x h g x h f x g x
hf x h f x g x h g x
h h
f x h f x g x h g x
h hf x g x
ProofSUM RULE
If f and g are both differentiable,
then
( ) ( ) ( ) ( )d d df x g x f x g x
dx dx dx
DIFFERENCE RULE
8 5 4 3
8 5 4
3
7 4 3 2
7 4 3 2
( 12 4 10 6 5)
12 4
10 6 5
8 12 5 4 4 10 3 6 1 0
8 60 16 30 6
dx x x x x
dxd d dx x x
dx dx dxd d dx x
dx dx dx
x x x x
x x x x
NEW DERIVATIVES FROM OLD Example 5
Find the points on the curve
y = x4 - 6x2 + 4
where the tangent line is horizontal.
NEW DERIVATIVES FROM OLD Example 6
Horizontal tangents occur where
the derivative is zero.
We have:
Thus, dy/dx = 0 if x = 0 or x2 – 3 = 0, that is, x = ± .
4 2
3 2
( ) 6 ( ) (4)
4 12 0 4 ( 3)
dy d d dx x
dx dx dx dx
x x x x
NEW DERIVATIVES FROM OLD Example 6
3
So, the given curve has horizontal tangents
when x = 0, , and - .
The corresponding points are (0, 4), ( , -5), and (- , -5).
NEW DERIVATIVES FROM OLD Example 6
33
3 3
The equation of motion of a particle is
s = 2t3 - 5t2 + 3t + 4, where s is measured
in centimeters and t in seconds.
Find the acceleration as a function of time.
What is the acceleration after 2 seconds?
NEW DERIVATIVES FROM OLD Example 7
The velocity and acceleration are:
The acceleration after 2 seconds is: a(2) = 14 cm/s2
2( ) 6 10 3
( ) 12 10
dsv t t t
dtdv
a t tdt
NEW DERIVATIVES FROM OLD Example 7
Let’s try to compute the derivative of
the exponential function f(x) = ax using
the definition of a derivative:
0 0
0 0
( ) ( )'( ) lim lim
( 1)lim lim
x h x
h h
x h x x h
h h
f x h f x a af x
h h
a a a a a
h h
EXPONENTIAL FUNCTIONS
The factor ax doesn’t depend on h.
So, we can take it in front of the limit:
0
1'( ) lim
hx
h
af x a
h
EXPONENTIAL FUNCTIONS
41
Calculating the derivative of g(x) = 2x
0
2 1lim ?
h
hBut what is
h
(2 )0.693 2 2
xx xdk
dx
: ln 2 .6931Note
42
Similar calculations show the following for other values of a:
( )x xd ak a
dx
: ln 2 .6931Note
ln 3 1.0986ln 4 1.3863ln 5 1.6094ln 6 1.7918ln 7 1.9459
Thus, we have:
(2 ) ln 2 2 (3 ) ln 3 3x x x xd d
dx dx
EXPONENTIAL FUNCTIONS Equation 5
Consider 0
1lim ,
h
h
e
h
DEFINITION OF THE NUMBER e
1
.001 .9995
.0001 .99995
.00001 .999995
.00001 1.000005
.0001 1.00005
.001 1.0005
heh
h
So it appears that .
0
1lim 1
h
h
e
h
Geometrically, this means that, of all the
possible exponential functions y = ax, the
function f(x) = ex is the one whose tangent line
at (0, 1) has a slope f’(0) that is exactly 1.
EXPONENTIAL FUNCTIONS
The derivative of the natural exponential
function is:
( )
( ) (ln )
x x
x x
de e
dx
da a a
dx
DERIV. OF NATURAL EXPONENTIAL FUNCTION
Thus, the exponential function f(x) = ex
has the property that it is its own derivative.
The geometrical significance of this fact is that the slope of a tangent line to the curve y = ex is equal to the y coordinate of the point.
EXPONENTIAL FUNCTIONS
If f(x) = ex - x, find f’ .
Compare the graphs of f and f’.
EXPONENTIAL FUNCTIONS Example 8
'( ) ( )
( ) ( )
1
x
x
x
df x e x
dxd de x
dx dx
e
The function f and its derivative f’ are
graphed here.
Notice that f has a horizontal tangent when x = 0.
This corresponds to the fact that f’(0) = 0.
EXPONENTIAL FUNCTIONS Example 8
Notice also that, for x > 0, f’(x) is positive
and f is increasing.
When x < 0, f’(x) is negative and f
is decreasing.
EXPONENTIAL FUNCTIONS Example 8