The product and quotient rules

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…

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 wher…

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 func…

Let $f$ and $g$ be differentiable functions for which the following information is known: $f(2) = 5$, $g(2) = -3$, $f'(2) = -1/2$, $g'(2) = 2$.

1. Let $h$ be the new function defined by the rule $h(x) = g(x) \cdot f(x)$. Determine $h(2)$ and $h'(2)$.

2. Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$ (where $h$ is the function defined in (a)).

3. Let $r$ be the function defined by the rule $r(x) = \frac{g(x)}{f(x)}$. Is $r$ increasing, decreasing, or neither at $a = 2$? Why?

4. Estimate the value of $r(2.06)$ (where $r$ is the fu…

Consider the functions $r(t) = t^t$ and $s(t) = \arccos(t)$, for which you are given the facts that $r'(t) = t^t(\ln(t) + 1)$ and $s'(t) = -\frac{1}{\sqrt{1-t^2}}$. Do not be concerned with where these derivative formulas come from. We restrict our interest in both functions to the domain $0 \lt t \lt 1$.

1. Let $w(t) = t^t \arccos(t)$. Determine $w'(t)$.

2. Find an equation for the tangent line to $y = w(t)$ at the point $(\frac{1}{2}, w(\frac{1}{2}))$.

3. Let $v(t) = \frac{t^t}{\arccos(t)}$. Is $v$ increasing or decreasing at the instant $t = \frac{1}{2}$? Why?

A farmer with large land holdings has historically grown a wide variety of crops. With the price of ethanol fuel rising, he decides that it would be prudent to devote more and more of his acreage to producing corn. As he grows more and more corn, he learns efficiencies that increase his yield per acre. In the present year, he used 7000 acres of his land to grow corn, and that land had an average y…

Let $f(v)$ be the gas consumption (in liters/km) of a car going at velocity $v$ (in km/hour). In other words, $f(v)$ tells you how many liters of gas the car uses to go one kilometer if it is traveling at $v$ kilometers per hour. In addition, suppose that $f(80)=0.05$ and $f'(80) = 0.0004$.

1. Let $g(v)$ be the distance the same car goes on one liter of gas at velocity $v$. What is the relationship between $f(v)$ and $g(v)$? Hence find $g(80)$ and $g'(80)$.

2. Let $h(v)$ be the g…