Activity 2.3.2

Use the product rule to respond to each of the prompts below. Throughout, be sure to carefully label any derivative you find by name. It is not necessary to algebraically simplify any of the derivatives you compute.

1. Let $m(w)=3w^{17} 4^w$. Find $m'(w)$.

2. Let $h(t) = (\sin(t) + \cos(t))t^4$. Find $h'(t)$.

3. Determine the slope of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ if $f$ is given by the rule $f(x) = e^x \sin(x)$.

4. Find the tangent line approximation $L(x)$ to the function $y = g(x)$ at the point $(-1,g(-1))$ if $g$ is given by the rule $g(x) = (x^2 + x) 2^x$.