The chain rule

For each function given below, identify an inner function $g$ and outer function $f$ to write the function in the form $f(g(x))$. Determine $f'(x)$, $g'(x)$, and $f'(g(x))$, and then apply the chain rule to determine the derivative of the given function.

1. $h(x) = \cos(x^4)$

2. $p(x) = \sqrt{ \tan(x) }$

3. $s(x) = 2^{\sin(x)}$

4. $z(x) = \cot^5(x)$

5. $m(x) = (\sec(x) + e^x)^9$

For each of the following functions, find the function's derivative. State the rule(s) you use, label relevant derivatives appropriately, and be sure to clearly identify your overall answer.

1. $p(r) = 4\sqrt{r^6 + 2e^r}$

2. $m(v) = \sin(v^2) \cos(v^3)$

3. $h(y) = \frac{\cos(10y)}{e^{4y}+1}$

4. $s(z) = 2^{z^2 \sec (z)}$

5. $c(x) = \sin(e^{x^2})$

Use known derivative rules, including the chain rule, as needed to respond to each of the following prompts.

1. Find an equation for the tangent line to the curve $y= \sqrt{e^x + 3}$ at the point where $x=0$.

2. If $\displaystyle s(t) = \frac{1}{(t^2+1)^3}$ represents the position function of a particle moving horizontally along an axis at time $t$ (where $s$ is measured in inches and $t$ in seconds), find the particle's instantaneous velocity at $t=1$. Is the particle moving…

Consider the basic functions $f(x) = x^3$ and $g(x) = \sin(x)$.

1. Let $h(x) = f(g(x))$. Find the exact instantaneous rate of change of $h$ at the point where $x = \frac{\pi}{4}$.

2. Which function is changing most rapidly at $x = 0.25$: $h(x) = f(g(x))$ or $r(x) = g(f(x))$? Why?

3. Let $h(x) = f(g(x))$ and $r(x) = g(f(x))$. Which of these functions has a derivative that is periodic? Why?

Let $u(x)$ be a differentiable function. For each of the following functions, determine the derivative. Each response will involve $u$ and/or $u'$.

1. $p(x) = e^{u(x)}$

2. $q(x) = u(e^x)$

3. $r(x) = \cot(u(x))$

4. $s(x) = u(\cot(x))$

5. $a(x) = u(x^4)$

6. $b(x) = (u(x))^4$

If a spherical tank of radius 4 feet has $h$ feet of water present in the tank, then the volume of water in the tank is given by the formula .

1. At what instantaneous rate is the volume of water in the tank changing with respect to the height of the water at the instant $h = 1$? What are the units on this quantity?

2. Now suppose that the height of water in the tank is being regulated by an inflow and outflow (e…