Section 3.1
Derivatives of Polynomials and
Exponential Functions

Goals
• Learn formulas for the derivatives of
Constant functions
 Power functions
 Exponential functions

• Learn to find new derivatives from old:
Constant multiples
 Sums and differences

Constant Functions

The graph of the constant function
f(x) = c is the horizontal line y = c
…
• which has slope 0 ,
• so we must have f(x) = 0 (see the
next slide).

A formal proof is easy:
Constant Functions (cont’d)
Power Functions
Next we look at the functions f(x) =
xn , where n is a positive integer.
 If n = 1 , then the graph of f(x) = x
is the line y = x , which has slope 1
, so f(x) = 1.
 We have already seen the cases n =
2 and n = 3 :

Power Functions (cont’d)

For n = 4 we find the derivative of
f(x) = x4 as follows:
Power Functions (cont’d)
There seems to be a pattern
emerging!
 It appears that in general, if f(x) =
xn , then

f(x) = nxn-1 .

This turns out to be the case:
Power Functions (cont’d)


We illustrate the Power Rule using a
variety of notations:
It turns out that the Power Rule is
valid for any real number n , not
just positive integers:
Power Functions (cont’d)
Constant Multiples

The following formula says that the
derivative of a constant times a function
is the constant times the derivative of
the function:
Sums and Differences

These next rules say that the
derivative of a sum (difference) of
functions is the sum (difference) of
the derivatives:
Example
Exponential Functions


If we try to use the definition of
derivative to find the derivative of
f(x) = ax , we get:
The factor ax doesn’t depend on x ,
so we can take it in front of the limit:
Exponential (cont’d)

Notice that the limit is the value of
the derivative of f at 0 , that is,
Exponential (cont’d)

This shows that…
• if the exponential function f(x) = ax is
differentiable at 0 ,
• then it is differentiable everywhere and
f(x) = f(0)ax

Thus, the rate of change of any
exponential function is proportional
to the function itself.
Exponential (cont’d)

The table shown gives numerical
evidence for the existence of f(0)
when
• a = 2 ; here apparently
f(0) ≈ 0.69
• a = 3 ; here apparently
f(0) ≈ 1.10
Exponential (cont’d)

So there should be a number a
between 2 and 3 for which f(0) =
1 , that is,
h
a 1
lim
1
h0
h
But the number e introduced in
Section 1.5 was chosen to have just
this property!
 This leads to the following definition:

Exponential (cont’d)

Geometrically, this means that
• of all the exponential functions y = ax ,
• the function f(x) = ex is the one whose
tangent at (0, 1) has a slope f(0)
that is exactly 1 .
• This is shown on the next slide:
Exponential (cont’d)
Exponential (cont’d)


This leads to the following
differentiation formula:
Thus, the exponential function f(x)
= ex is its own derivative.
Example
If f(x) = ex – x , find f(x) and
f(0) .
 Solution The Difference Rule gives


Therefore
Solution (cont’d)
Note that ex is positive for all x , so
f(x) > 0 for all x .
 Thus, the graph of
f is concave up.

• This is confirmed
by the graph shown.
Review
Derivative formulas for polynomial
and exponential functions
 Sum and Difference Rules
 The natural exponential function ex
