f  x   b , b  0  b  1
x
• Exponential functions are positive: bx > 0 for all x.
• The range of f (x) = bx is the set of all positive real numbers.
• f (x) = bx is increasing if b > 1 and decreasing if 0 < b < 1.
f (x) = bx increases more rapidly than the power function xn for all n
Comparison of 3x and power functions
Laws of Exponents (b > 0)
Rewrite as a whole number or fraction:
(a) 161/ 2
(b) 27 2/3 (c) 416  418
93
(d) 7
3
Solve for the unknown:
(a) 2
3 x 1
2
5
3x  1  5
4
x
3
(b) b  5
3
6
b  5

6 1/ 3
b  25
(c) 7
t 1
1
 
7
2t
7
t 1
 7

1 2 t
t  1  2t
1
3t  1  t  
3
At this point, it is natural to ask: What is the derivative of
f (x) = bx? Our rules of differentiation are of no help because bx is
neither a product, quotient, nor composite of functions with known
derivatives. We must go back to the limit definition of the
derivative. The difference quotient (for h  0) is
b x  b h  1
f  x  h  f  x b  b
b b b



h
h
h
h
x
h
b
b
 1

d x
x
 b  lim
The
factor
b
doesn't depend on h 

h 0
dx
h
h
h



d x
b

1
b
1 
x
 b  b lim 
 Let m  b   lim 


h 0
h 0
dx
h
h




xh
d x
 b  m b bx
dx
x
x h
x
What is the factor m  b  ?
At this point, it is natural to ask: What is the derivative of
f (x) = bx? Our rules of differentiation are of no help because bx is
neither a product, quotient, nor composite of functions with known
derivatives. We must go back to the limit definition of the
derivative. The difference quotient (for h  0) is
 bh  1 
m  b   lim 

h 0
h


We cannot determine its exact value at this point (in Section 7.3, we
will learn that m (b) is equal to ln b, the natural logarithm of b). To
proceed further, let’s investigate m (b) numerically.
Estimate m (b) numerically for b = 2, 2.5, 3, and 10.
These computations suggest that m (b) is an increasing function of b.
In fact, it can be shown that m (b) is both increasing and continuous
as a function of b (we shall take these facts for granted). Then, since
m (2.5) ≈ 0.92 and m (3) ≈ 1.10, there exists a unique number b
between 2.5 and 3 such that m (b) = 1. This is the number e, whose
value is approximately 2.718.
Using infinite series (Exercise 87 in Section 11.7), we can show that
e is irrational and we can compute its value to any desired accuracy.
Because e is defined by the property m (e) = 1, Eq. (1) tells us
that (ex)' = ex. In other words, ex is equal to its own derivative.
Estimate m (b) numerically for b = 2, 2.5, 3, and 10.
The Number e
the property:
There is a unique positive real number e with
d x
e  ex
dx
The number e is irrational, with approximate value e ≈ 2.718.
Whenever we refer to the exponential function without
specifying the base, the reference is to f (x) = ex. In many
books, ex is denoted exp(x).
GRAPHICAL INSIGHT The graph of f (x) = bx passes through
(0, 1) for all b > 0 because b0 = 1. The number m (b) is simply the
slope of the tangent line at x = 0:
These tangent lines become steeper as b increases
and b = e is the unique value for which the
tangent line has slope 1. In Section 7.3, we will
show that m (b) = ln b, the natural logarithm of b.
CONCEPTUAL INSIGHT In some ways, the number e is
“complicated”. It has been computed to an accuracy of more than 100
billion digits, but it is irrational and it cannot be defined without using
limits. To 20 places, e = 2.71828182845904523536…
d x
e  ex
However, the elegant formula
dx
shows that e is “simple” from the point of view of calculus and that ex
is simpler than the seemingly more natural exponential functions
such as 2x or 10x.
Although written reference to the number π
goes back more than 4000 years,
mathematicians first became aware of the
special role played by e in the seventeenth
century. The notation e was introduced
around 1730 by Leonhard Euler, who
discovered many fundamental properties of
this important number.
Find the equation of the tangent line to the graph of f (x) = 3ex − 5x2 at
x = 2.
Calculate f '  0  , where f  x   e x cos x.
Differentiate: (a) f  x   e9 x 5 (b) f  x   e cos x
Graph Sketching Involving ex Sketch the graph of f (x) = xex on the
interval [−4, 2].
f '  x   e  xe  e 1  x   c  1
x
x
 0, x 1
 0, x 1
 f  1 is a local min.
f ' x  {
x
f "  x   e 1  x   e  2e  xe  e  2  x   c  2
x
f ''  x   {
 0, x 2
 0, x 2
x
x
x
 f  2  is an IP
lim f  x   0 lim f  x   
x 
x 
x
u  7 x  5  du  7dx
e

7 x  5
dx  ?
1 u
1 u
  e du   e  C 
7
7
1 7 x 5
 e
C
7
u  2 x  du  4 xdx
2
xe
dx

?

2 x2
1 u
1 u


e
du

e

C



4
4
1 2 x2
 e C
4
t
e
dt

?
 1  2et  e2t
u  e  du  e dt
t
du
du
 1  2u  u 2   1  u 2
1
t
z  1  u  dz  du
z
1
1
  z dz 
  1  u   
C
t
1
1 e
2
CONCEPTUAL INSIGHT What precisely do we mean by bx? We
have taken for granted that bx is meaningful for all real numbers x, but
we never specified how bx is defined when x is irrational. If n is a
whole number, then bn is simply the product b · b ··· b (n times), and
for any rational number x = m/n,
b b
x
m/n
 b 
n
m
 b
n
m
When x is irrational, this definition does not apply and bx cannot be
defined directly in terms of roots and powers of b. However, it makes
sense to view bm/n as an approximation to bx when m/n is a rational
should be approximately equal
number close to x. For example,
to 31.4142 ≈ 4.729 because 1.4142 is a good rational approximation to
. Formally, then, we may define bx as a limit over rational numbers
m/n approaching x:
x
m/n
b  lim b
m / n x
It can be shown that this limit exists and that the function f (x) = bx
(thus defined) is not only continuous, but also differentiable.