If the expression for g(x) is first mulitplied out, showing that g(x) = x6-1, the same result follows immediately.
Then, using the product rule again, and supressing the dependence on the variable x,
and substitution for k(x) in terms of g(x) and h(x) gives the result.
Letting g(x) = h(x) = f(x) leads to the corollary
The result of the above exercise gives
and it is quite questionable whether the last expansion is worth the effort; in fact, expanding the original function and then differentiating should lead to the same result. You are encouraged to check this.