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.

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Last modified August 28, 1998