Procedure

As previously noted, the given function f(t) = 4t sin(xt) is the product of two differentiable functions of the form f(t) = g(t)h(t), where g(t) = 4t and h(t) = sin(nt). We determined that h'(t) = n cos(nt). So the last thing we must do before applying the product rule is to find g'(t). g(t) = 4t g'(t) We now apply the product rule. g(t)h(t) )' = g(t)h'(t) + h(t)g'(t) = 4t(n cos(nt)) + sin(nt) Therefore, we have the following result. f' (t) =