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
