You can prove that for all using the chain rule.
.
Now let so that .
.
.
Thus
.
Q.E.D.
Of course, you will need to have proved the chain rule and the derivatives of and beforehand...
such that if
then
using the definition of a derivative (the 'lim as h tends to zero' one), the formula
and fact that the derivative of is (proved earlier using the above formula, where the "h" falls out nicely)
I have achieved the result implicitly, but did not use the def-n of the derivative or the formula