Section 5.4
Exponential Functions:
Differentiation and Integration
Section 5.4
Exponential Functions: Differentiation and
Integration

Since we know how to deal with the
natural logarithm function for integrating
and differentiating, and we have
refreshed ourselves on the issue of
inverse functions, it stands to reason
that we can discuss the derivative and
antiderivative of the natural exponential
function
ye
x
Section 5.4
Exponential Functions: Differentiation and
Integration

Remember the following relationship:
x
x  ln y
ye
if and only if
 Since we know how to take the
derivative of a natural logarithm function,
we should transform the equations above
as follows:
d
1 dy
dy
dy
x
y  e  x  ln y 
y
e 
 x  ln y   1 
dx
y dx
dx
dx
x
Section 5.4
Exponential Functions: Differentiation and
Integration

So, what we have with the natural
exponential function is a function that
is its own derivative. The chain rule
leads us to a relatively logical extension
that looks like this, with two examples
below:
dy
du
y  eu 
dx
 eu
dx
dy
dy
3x
sin 3 x
ye 
 3e , y  e

 3cos 3xesin 3 x
dx
dx
3x
Section 5.4
Exponential Functions: Differentiation and
Integration

How can we deal with integrals involving
functions with a base of e? We need to
keep the chain rule in mind at all times
with these problems. We will be looking
to divide the derivative of the power of
the exponential function in each case.
Look at the following example:
1 5x
5x
 e dx  5 e  C
Section 5.4
Exponential Functions: Differentiation and
Integration

What if the exponent is more complicated
and has a function as its derivative instead
of simply a constant? In that case we need
to have a product of functions involved in
the integral. Look at the two examples
below, one of which we can integrate.
Decide which one is doable at this point.
 xe
x2
dx
 2e
x2
dx
Section 5.4
Exponential Functions: Differentiation and
Integration

Let’s try a couple of examples of definite
integrals. Remember, look for
substitutions so that you can see the
u du
function in the form of e
dx
ln 5

1
2
3x
e dx
 xe
0
 x2
dx
Section 5.4
Exponential Functions: Differentiation and
Integration

A couple of more examples…
e

1
2x
dx
2
x
3e3 x
 5  e3 x dx