Activity 2.3.4

Use relevant derivative rules to respond to each of the prompts below. Throughout, be sure to use proper notation and carefully label any derivative you find by name.

1. Let $f(r) = (5r^3 + \sin(r))(4^r - 2\cos(r))$. Find $f'(r)$.

2. Let $\displaystyle p(t) = \frac{\cos(t)}{t^6 \cdot 6^t}$. Find $p'(t)$.

3. Let $g(z) = 3z^7 e^z - 2z^2 \sin(z) + \frac{z}{z^2 + 1}$. Find $g'(z)$.

4. A moving particle has its position in feet at time $t$ in seconds given by the function $s(t) = \frac{3\cos(t) - \sin(t)}{e^t}$. Find the particle's instantaneous velocity at the moment $t = 1$.

5. Suppose that $f(x)$ and $g(x)$ are differentiable functions and it is known that $f(3) = -2$, $f'(3) = 7$, $g(3) = 4$, and $g'(3) = -1$. If $p(x) = f(x) \cdot g(x)$ and $\displaystyle q(x) = \frac{f(x)}{g(x)}$, calculate $p'(3)$ and $q'(3)$.