Proof of the Product Rule of Differentiation

Assume the functions are differentiable on suitable open intervals, with the composite argument in the domain of the outer function.

Let \({\mathrm{h}(x) = \mathrm{f}(x)\mathrm{g}(x)}\). Show that \({\mathrm{h}'(x) = \mathrm{f}'(x)\mathrm{g}(x) + \mathrm{f}(x)\mathrm{g}'(x)}\).