Derivative, by first principle

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