4.4
Exponential Functions
OBJECTIVES


Differentiate exponential functions.
Solve application problems with
exponential functions.
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DEFINITION:
An exponential function f is given by
f (x)  a x ,
where x is any real number, a > 0, and a ≠ 1. The
number a is called the base.
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Example 1: Look at the graph
function values.
First, we find some
f (x)  2 x.
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DEFINITION:
e is a number, named for the Swiss mathematician
Leonhard Euler.
e  lim 1 h   2.718281828459
1h
h0
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THEOREM 1
d x
x
e e
dx
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Example 1: Find dy/dx:
a) y  3e ;
x
b) y  x e ;
2 x
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x
e
c) y  3 .
x
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dy
x
a)
3e
dx
 
d 2 x
b)
x e
dx


d x
 3 e
dx
x

3e

x  e  e  2x

e x  2x
2
x
x

x
2
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
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Example 1 (concluded):
x

d e 
c)
dx  x 3 



x  e  e  3x
3
x
x
2
x 
3 2
x e x  3
x6
e x (x  3)
4
x
2 x
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THEOREM 2
d f (x)
f (x)
e  e  f (x)
dx
OR
d u
u du
e e 
dx
dx
The derivative of e to some power is the product of e
to that power and the derivative of the power.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example 2: Differentiate each of the following with
respect to x:
a) y  e ;
b) y  e
8x
c) y  e
x 2 3
 x 2  4 x 7
;
.
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d 8x
a)
e
dx
 e 8
8x

d  x 2 4 x7
b)
e
dx
8e
8x
 e
 x 2 4 x7
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 2x  4 
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Example 2 (concluded):
d
c)
e
dx
x 2 3
d  x 2 3 
e
dx

 e

1
2
 x 3  1
2

2
1
2
1
2
2
x
  3  2 x

x 2 3
xe
x 3
2
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The sales of a new computer ( in thousands) are
given by: S (t )  100  90e0.3t
t represents time in years.
Find the rate of change of sales at each time.
a) after 1 year
b) after 5 years
c) What is happening to the rate of change of sales?
Answers: a) 20 b) 6 c) decreasing
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Write an equation of the tangent line to
f ( x)  2e
3 x
at x = 0.
Answer: y = -6x+2
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