§2.5 The chain rule (continued)

Using multiple rules simultaneously

The composite version of basic function rules

Summary

For $h(t) = 3^{t^2+2t}\sec^4(t)$, which combination of rules does the derivative require?

For $p(r) = 4\sqrt{r^6 + 2e^r}$, which expression is $p'(r)$?

For $m(v) = \sin(v^2)\cos(v^3)$, which structure does $m'(v)$ have?

For $h(y) = \frac{\cos(10y)}{e^{4y}+1}$, which structure does $h'(y)$ have?

For $s(z) = 2^{z^2\sec(z)}$, which structure does $s'(z)$ have?

For $c(x) = \sin(e^{x^2})$, which expression is $c'(x)$?

Suppose $f(-1) = 2$, $f'(-1) = -5$, $f(2) = -3$, $f'(2) = 4$, $g(-1) = -3$, $g'(-1) = 4$, $g(2) = -1$, and $g'(2) = 2$. For $C(x) = f(g(x))$, compute $C'(2)$.

With the same table, for $D(x) = f(f(x))$ compute $D'(-1)$, using $D'(x) = f'(f(x)) \cdot f'(x)$.

For composite functions with linear inside, such as $\tan(17x)$, the chain rule gives

The chain rule upgrades $\frac{d}{dx}[\sin(x)] = \cos(x)$ and $\frac{d}{dx}[a^x] = a^x\ln(a)$ to which general statements?

With $f(x) = x^3$ and $g(x) = \sin(x)$, the derivatives satisfy $h'(0.25) \approx 0.178$ for $h = f(g(x))$ and $r'(0.25) \approx 0.188$ for $r = g(f(x))$. Also, which derivative is periodic?

Let $u$ be differentiable. Which list correctly pairs each function with its derivative?

Piecewise linear $p$ and $q$ satisfy $q(0) = 2$, $p'(2) = 2$, $q'(0) = 0$, $q(3) = 0$, $p'(0) = \frac{1}{2}$, and $q'(3) = -1$. For $C(x) = p(q(x))$, the values $C'(0)$ and $C'(3)$ are

With the same graphs, $Y(x) = q(q(x))$ has no derivative at $x = -2$, because $q(-2) = -1$ and $q$ has a corner at $-1$.

Let $u$ be differentiable. For $q(x) = u(e^x)$, which expression is $q'(x)$?

A spherical tank has $V = \frac{\pi}{3}h^2(12-h)$ cubic feet at water height $h$ feet, and the height follows $h(t) = \sin(\pi t) + 1$ feet at $t$ hours. At $t = 2$, the height satisfies $h(2) = 1$ and $h'(2) = \pi$. The rate $\frac{dV}{dt}$ at $t = 2$ equals

Water volume in a spherical tank is $V = \frac{\pi}{3}h^2(12-h)$ cubic feet at height $h$ feet. Compute $\frac{dV}{dh}$ at $h = 1$ rounded to three decimals.

In the tank problem, $\frac{dV}{dh}$ at $h = 1$ measures volume change per extra foot of height, while $\frac{dV}{dt}$ at $t = 2$ measures volume change per extra hour, and the two are linked by the height's own rate of change.

Air pressure is $P = 30e^{-0.0000323h}$ inches of mercury at altitude $h$ feet. Compute $P'(1000)$ rounded to six decimals.

The value $P'(1000) \approx -0.000938$ inches of mercury per foot means the pressure drops by about that amount for each additional foot of elevation near $1000$ feet.

The tangent line to $y = \sqrt{e^x + 3}$ at $x = 0$ has slope $\frac{1}{4}$.

A particle has position $s(t) = \frac{1}{(t^2+1)^3}$ inches at $t$ seconds. Compute the velocity $s'(1)$ as a fraction.