Derivatives of Exponential Functions

Let .  We will resort to the definition of the derivative in order to find  in this case.

Although this is not a limit we have seen before, it is related to a limit we have seen before –  – and we can use it to help us.  This limit tells us that for sufficiently small values of h we can say that .  If we manipulate this expression, we wil get what we need to evaluate our limit.

Therefore,  and we can finish finding our derivative.

So we now have that the derivative of  is ; that is, .  We shall see later that this is a defining characteristic for this function.

Now, what about the general exponential function?  We can now handle that with a simple application of the Chain Rule as follows.  Assume  and .

Now we apply the Chain Rule to this last form of the function.

In that last step, we just substituted our original form of the function for the last form.  So we now have that the derivative of  is .