The next set of functions that we want to take a look at are
exponential and logarithm functions. The
most common exponential and logarithm functions in a calculus course are the
natural exponential function, ,
and the natural logarithm function, . We will take a more general approach however
and look at the general exponential and logarithm function.
We’ll start off by looking at the exponential function,
We want to differentiate this. The power rule that we looked at a couple of
sections ago won’t work as that required the exponent to be a fixed number and
the base to be a variable. That is
exactly the opposite from what we’ve got with this function. So, we’re going to have to start with the
definition of the derivative.
Now, the is not affected by the limit since it
doesn’t have any h’s in it and so is
a constant as far as the limit is concerned.
We can therefore factor this out of the limit. This gives,
Now let’s notice that the limit we’ve got above is exactly
the definition of the derivative of at ,
i.e. . Therefore, the derivative becomes,
So, we are kind of stuck. We need to know the derivative in
order to get the derivative!
There is one value of a
that we can deal with at this point.
Back in the Exponential Functions section
of the Review chapter we stated that What we didn’t do however is actually define
where e comes from. There are in fact a variety of ways to define
Here are three of them.
Some Definitions of e.
- e is the unique positive number for which
The second one is the important one for us because that
limit is exactly the limit that we’re working with above. So, this definition leads to the following
So, provided we are using the natural exponential function
we get the following.
At this point we’re missing some knowledge that will allow
us to easily get the derivative for a general function. Eventually
we will be able to show that for a general exponential function we have,
Let’s now briefly get the derivatives for logarithms. In this case we will need to start with the
following fact about functions that are inverses of each other.
If f(x) and g(x) are inverses of each other then,
So, how is this fact useful to us? Well recall
that the natural exponential function and the natural logarithm function are
inverses of each other and we know what the derivative of the natural
exponential function is!
So, if we have and then,
The last step just uses the fact that the two functions are
inverses of each other.
Putting this all together gives,
Note that we need to require that since this is required for the logarithm and
so must also be required for its derivative.
It can also be shown that,
Using this all we need to avoid is .
In this case, unlike the exponential function case, we can
actually find the derivative of the general logarithm function. All that we need is the derivative of the
natural logarithm, which we just found, and the change of base formula. Using the change of base formula we can write
a general logarithm as,
Differentiation is then fairly simple.
We took advantage of the fact that a was a constant and so is also a constant and can be factored out of
the derivative. Putting all this
Here is a summary of the derivatives in this section.
Okay, now that we have the derivations of the formulas out
of the way let’s compute a couple of derivatives.
Example 1 Differentiate
each of the following functions.
(a) This will
be the only example that doesn’t involve the natural exponential and natural
(b) Not much to
this one. Just remember to use the
product rule on the second term.
(c) We’ll need
to use the quotient rule on this one.
There’s really not a lot to differentiating natural
logarithms and natural exponential functions at this point as long as you
remember the formulas. In later sections
as we get more formulas under our belt they will become more complicated.
Next, we need to do our obligatory
application/interpretation problem so we don’t forget about them.
Example 2 Suppose
that the position of an object is given by
Does the object ever stop moving?
First we will need the derivative. We need this to determine if the object
ever stops moving since at that point (provided there is one) the velocity
will be zero and recall that the derivative of the position function is the
velocity of the object.
The derivative is,
So, we need to determine if the derivative is ever
zero. To do this we will need to
Now, we know that exponential functions are never zero and
so this will only be zero at . So, if we are going to allow negative
values of t then the object will
stop moving once at . If we aren’t going to allow negative values
of t then the object will never
Before moving on to the next section we need to go back over
a couple of derivatives to make sure that we don’t confuse the two. The two derivatives are,
It is important to note that with the Power rule the exponent
MUST be a constant and the base MUST be a variable while we need exactly the
opposite for the derivative of an exponential function. For an exponential function the exponent MUST
be a variable and the base MUST be a constant.
It is easy to get locked into one of these formulas and just
use it for both of these. We also
haven’t even talked about what to do if both the exponent and the base involve
variables. We’ll see this situation in a