
Employ a standard trick used in proving the chain rule:

The limit of a product is the product of limits, i.e. we can write

The rightmost limit is an exercise in differentiating 

 using the definition, which you probably already know is 

.
For the leftmost limit, we make a substitution 

. Now, if we make a slight change to 

 by adding a small number 

, this propagates a similar small change in 

 that we'll call 

, so that we can set 

. Then as 

, we see that it's also the case that 

 (since we fix 

). So we can write the remaining limit as

which in turn is the derivative of 

, another limit you probably already know how to compute. We'd end up with 

, or 

.
So we find that
