The Product Rule: Solutions

1. 
   

   leading to

   

   If the expression for g(x) is first mulitplied out, showing that g(x) = x⁶-1, the same result follows immediately.

2. Let k(x)  = g(x)h(x), so that
   

   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

   

3. Given:

   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.