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.
A moving particle has its position in feet at time \(t\) in seconds given by the function \(s(t) = \frac{3\cos(t) - \sin(t)}{e^t}\text{.}\) Find the particle’s instantaneous velocity at the moment \(t = 1\text{.}\)
Suppose that \(f(x)\) and \(g(x)\) are differentiable functions and it is known that \(f(3) = -2\text{,}\)\(f'(3) = 7\text{,}\)\(g(3) = 4\text{,}\) and \(g'(3) = -1\text{.}\) If \(p(x) = f(x) \cdot g(x)\) and \(\displaystyle q(x) = \frac{f(x)}{g(x)}\text{,}\) calculate \(p'(3)\) and \(q'(3)\text{.}\)
