Activity 2.3.3

Use the quotient rule to respond to each of the prompts below. Throughout, be sure to carefully label any derivative you find by name. That is, if you're given a formula for $f(x)$, clearly label the formula you find for $f'(x)$. It is not necessary to algebraically simplify any of the derivatives you compute.

1. Let $r(z)=\frac{3^z}{z^4 + 1}$. Find $r'(z)$.

2. Let $v(t) = \frac{\sin(t)}{\cos(t) + t^2}$. Find $v'(t)$.

3. Determine the slope of the tangent line to the curve $\displaystyle R(x) = \frac{x^2 - 2x - 8}{x^2 - 9}$ at the point where $x = 0$.

4. When a camera flashes, the intensity $I$ of light seen by the eye is given by the function $$\begin{gathered} I(t) = \frac{100t}{e^t} \end{gathered}$$, where $I$ is measured in candles and $t$ is measured in milliseconds. Compute $I'(0.5)$, $I'(2)$, and $I'(5)$; include appropriate units on each value; and discuss the meaning of each.