Previously, we developed the Product Rule and studied how it is employed to
differentiate a product of two functions. In particular, recall that if $f$ and $g$ are
differentiable functions of $x$, then
$\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot g'(x) + g(x) \cdot f'(x)$.
(a) For each of the following functions, use the Product Rule to find the function's
derivative. Notice the label of the derivative (e.g., the derivative of $g(x)$ should be
labeled $g'(x)$).
(i) If $g(x) = x \sin(x)$, then $g'(x) = x\cos x + \sin x$
(ii) If $h(x) = xe^x$, then
(iii) If $p(x) = x\ln(x)$, then
(iv.) If $q(x) = x^2 \cos(x)$, then
(v.) If $r(x) = e^x \sin(x)$, then
