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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
However, it would be tedious if we always had
to use the definition.
So, in this chapter, we develop rules for finding
derivatives without having to use the definition
directly.
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
3.1
Derivatives 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 us 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 lim 0 0
h
h h
f x h f xf x
hc c
h
CONSTANT FUNCTION
In Leibniz notation, we write this rule as follows.
( ) 0d
cdx
DERIVATIVE
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.
( ) 1d
xdx
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 ( ) 3 d d
x x x xdx 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
Thus,
4 3( ) 4d
x xdx
POWER FUNCTIONS Equation 3
Comparing Equations 1, 2, and 3, we see a
pattern emerging.
It seems to be a reasonable guess that, when n is a positive integer, (d/dx)(xn) = nxn - 1.
This turns out to be true.
POWER FUNCTIONS
If n is a positive integer, then
1( ) n ndx nx
dx
POWER RULE
The formula
can be verified simply by multiplying out the
right-hand side (or by summing the second
factor as a geometric series).
1 2 2 1( )( ) n n n n n nx a x a x x a xa a
Proof 1POWER RULE
If f (x) = x n, we can use Equation 5 in Section
2.7 for f’(a) and the previous equation to write:
1 2 2 1
1 2 2 1
1
( ) ( )'( ) lim lim
lim( )
n n
x a x a
n n n n
x a
n n n n
n
f x f a x af a
x a x a
x x a xa a
a a a aa a
na
POWER RULE Proof 1
We illustrate the Power Rule using various
notations in Example 1.
POWER RULE
a.If f (x) = x6, then f’(x) = 6x5
b.If y = x1000, then y’ = 1000x999
c.If y = t 4, then
d. 3 2( ) 3
dr r
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 1
d
dx x x
1 21 d
x xdx
NEGATIVE INTEGERS
So, the Power Rule is true when n = -1.
In fact, we will show in the next section that it holds for all negative integers.
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
d
x xdx
FRACTIONS
This shows that the Power Rule is true even
when n = ½.
In fact, we will show in Section 3.6 that it is true
for all real numbers n.
FRACTIONS
If n is any real number, then
1( ) n ndx nx
dx
POWER RULE—GENERAL VERSION
Differentiate:
a.
b.
In each case, we rewrite the function as a power of x.
2
1( ) f x
x3 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 dxd
xdx
x x
POWER RULE Example 2 b
The Power Rule enables us to find tangent lines
without having to resort to the definition of a
derivative.
TANGENT LINES
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 2 y 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:
52 23 3 31 ( 1) or y x y x
Example 3NORMAL LINE
322
3
We graph the curve and its tangent line and
normal line here.
Example 3TANGENT AND NORMAL LINES
When new functions are formed from old
functions by addition, subtraction, or
multiplication by a constant, their derivatives
can be calculated in terms of derivatives of the
old functions.
NEW DERIVATIVES FROM OLD
In particular, the following formula says that:
the derivative of a constant times a function
is the constant times the derivative of the
function.
NEW DERIVATIVES FROM OLD
If c is a constant and f is a differentiable
function, then
( ) ( )d d
cf x c f xdx 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 dxd
xdx
NEW DERIVATIVES FROM OLD Example 4
The next rule tells us that the derivative of a sum
of functions is the sum of the derivatives.
NEW DERIVATIVES FROM OLD
If f and g are both differentiable, then
( ) ( ) ( ) ( ) d d d
f x g x f x g xdx 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
The Sum Rule can be extended to the sum of any
number of functions.
For instance, using this theorem twice, we get:
( ) ( )
( )
f g h f g h
f g h
f g h
SUM RULE
By writing f - g as f + (-1)g and applying the
Sum Rule and the Constant Multiple Rule, we
get the following formula.
NEW DERIVATIVES FROM OLD
If f and g are both differentiable, then
( ) ( ) ( ) ( ) d d d
f x g x f x g xdx dx dx
DIFFERENCE RULE
The Constant Multiple Rule, the Sum Rule, and
the Difference Rule can be combined with the
Power Rule to differentiate any polynomial—as
the following examples demonstrate.
NEW DERIVATIVES FROM OLD
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 d
x x xdx dx dx
d d dx x
dx dx dx
x x x x
x x x x
NEW DERIVATIVES FROM OLD Example 5
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
dx
x x x x x
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,
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
3x
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 us try to compute the derivative of the
exponential function f(x) = a x 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 does not depend on h. So, we can
take it in front of the limit:
Notice that the limit is the value of the derivative of f at 0, that is,
0
1( ) lim
h
x
h
af x a
h
0
1lim (0)
h
h
af
h
EXPONENTIAL FUNCTIONS
The factor ax does not depend on h. So, we can
take it in front of the limit:
Notice that the limit is the value of the derivative of f at 0, that is,
0
1( ) lim
h
x
h
af x a
h
0
1lim ln
h
h
aa
h
EXPONENTIAL FUNCTIONS
Thus, we have shown that, if the exponential
function f (x) = a x is differentiable at 0, then
it is differentiable everywhere and
The equation states that the rate of change of any exponential function is proportional to the function itself.
The slope is proportional to the height.
EXPONENTIAL FUNCTIONS Equation 4
( ) (0) ln x xf x f a a a
If we put a = e and, therefore, f’(0) = ln(e) = 1 in
Equation 4, it becomes the following important
differentiation formula.
EXPONENTIAL FUNCTIONS
The derivative of the natural exponential
function is:
( ) x xde e
dx
DERIVATIVE OF THE NATURAL EXPONENTIAL FUNCTION
Thus, the exponential function f(x) = e x 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 coordinate of the point.
EXPONENTIAL FUNCTIONS
If f(x) = ex - x, find f’ and f’’.
Compare the graphs of f and f’.
EXPONENTIAL FUNCTIONS Example 8
Using the Difference Rule, we have:
( ) ( )
( ) ( )
1
x
x
x
df x e x
dxd d
e xdx dx
e
EXPONENTIAL FUNCTIONS Example 8
In Section 2.8, we defined the second derivative
as the derivative of f’.
So,( ) ( 1)
( ) (1)
x
x
x
df x e
dxd d
edx dx
e
EXPONENTIAL FUNCTIONS Example 8
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
At what point on the curve y = ex is the tangent
line parallel to the line y = 2x?
EXPONENTIAL FUNCTIONS Example 9
Since y = ex, we have y’ = ex.
Let the x-coordinate of the point in question be
a.
Then, the slope of the tangent line at that point is ea.
EXPONENTIAL FUNCTIONS Example 9
This tangent line will be parallel to the line
y = 2x if it has the same slope, that is, 2.
Equating slopes, we get:
ea = 2 a = ln 2
EXPONENTIAL FUNCTIONS Example 9
Thus, the required point is: (a, ea) = (ln 2, 2)
EXPONENTIAL FUNCTIONS Example 9