Derivatives of Exponential and Logarithm Functions
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.
Exponential Functions
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 at 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 do actually define where e comes from. There are in
fact a variety of ways to define e. Here are a three of them.
Some Definitions of e.
 - e is the unique positive number for
which
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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 fact,
Fact 1
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For the natural exponential function,
 we
have
 . |
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,
Logarithm Functions
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.
Fact 2
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If f(x) and g(x) are inverses of each
other then,
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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. In 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
together gives,
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.
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