We first observe that
is the product of two functions:
where
and
We will need to use the product rule to differentiate
And because
and
are composite functions, we will need the chain rule. We therefore begin by computing
and
Writing
and finding the derivatives of
and
we have
Thus, by the chain rule, it follows that
Turning next to
we write
and find the derivatives of
and
Now we are finally ready to compute the derivative of the function Recalling that by the product rule we have
From our work above with and we know the derivatives of and and therefore