We derive the product rule for derivatives to show why when we differentiate the product function
 
 

the result is more complex than the product of the derivatives of the functions and.  (See example below.)
 

So, we begin by taking the formal derivative of the function 
 
 

which becomes
 
 
 

when we rewrite as the product of and .
 

Now, we will be unable to resolve the limit any further unless we apply a clever trick--to add and subtract the same quantity to the numerator of the limit--so that we may then separate the fraction into two resolvable limits.

So, we rewrite as
 
 

and then split the limit into two parts, as follows
 
 

The derivation is completed by noticing that
 
 

and that
 
 

So we obtain the result
 
 
 

What follows is an example to show why the derivative of the product function is of the form above and why  is  not
 
 

Let
 
 

Note the we can take the derivative of most easily (without using the product rule) by rewriting as
 
 

and then by taking the derivative of this using the power rule for derivatives to obtain
 
 

Now, suppose we see what happens when we make the incorrect assumption that the derivative of is
 
 

We then would have that
 
 

which we know is incorrect.  However if we apply the product rule that
 
 
 

we obtain the correct result