A table gives: at $x=1$, $f=3,f^{\prime }=2,g=4,g^{\prime }=-1$; at $x=2$, $f=1,f^{\prime }=5,g=2,g^{\prime }=3$. If $h(x)=f(x)g(x)$, find $h^{\prime }(1)$.

A table gives: at $x=1$, $f=2,f^{\prime }=1,g=5,g^{\prime }=4$; at $x=2$, $f=6,f^{\prime }=3,g=2,g^{\prime }=-2$. If $q(x)=\frac{f(x)}{g(x)}$, find $q^{\prime }(2)$.

A table gives: at $x=0$, $f=-1,f^{\prime }=4,g=3,g^{\prime }=2$; at $x=1$, $f=2,f^{\prime }=1,g=5,g^{\prime }=-3$. If $h(x)=f(x)g(x)$, find $h^{\prime }(0)$.

A table gives: at $x=1$, $f=2,f^{\prime }=4,g=1,g^{\prime }=2$; at $x=3$, $f=4,f^{\prime }=-3,g=2,g^{\prime }=5$. If $q(x)=\frac{f(x)}{g(x)}$, find $q^{\prime }(3)$.

A table gives: at $x=2$, $f=2,f^{\prime }=-1,g=-3,g^{\prime }=4$; at $x=4$, $f=1,f^{\prime }=2,g=5,g^{\prime }=1$. If $h(x)=f(x)g(x)$, find $h^{\prime }(2)$.

A table gives: at $x=1$, $f=3,f^{\prime }=2,g=4,g^{\prime }=1$; at $x=2$, $f=5,f^{\prime }=6,g=-1,g^{\prime }=3$. If $q(x)=\frac{f(x)}{g(x)}$, find $q^{\prime }(2)$.

A table gives: at $x=3$, $f=0,f^{\prime }=7,g=4,g^{\prime }=-2$; at $x=4$, $f=2,f^{\prime }=1,g=1,g^{\prime }=5$. If $h(x)=f(x)g(x)$, find $h^{\prime }(3)$.

A table gives: at $x=1$, $f=4,f^{\prime }=2,g=2,g^{\prime }=1$; at $x=2$, $f=-2,f^{\prime }=5,g=3,g^{\prime }=-1$. If $q(x)=\frac{f(x)}{g(x)}$, find $q^{\prime }(2)$.

A table gives: at $x=1$, $f=2,f^{\prime }=5,g=3,g^{\prime }=-2$; at $x=2$, $f=7,f^{\prime }=-1,g=1,g^{\prime }=4$; at $x=3$, $f=6,f^{\prime }=4,g=2,g^{\prime }=3$. If $k(x)=f(g(x))$, find $k^{\prime }(1)$.

A table gives: at $x=2$, $f=1,f^{\prime }=4,g=4,g^{\prime }=3$; at $x=4$, $f=8,f^{\prime }=-2,g=1,g^{\prime }=-1$; at $x=5$, $f=3,f^{\prime }=2,g=2,g^{\prime }=1$. If $k(x)=f(g(x))$, find $k^{\prime }(2)$.

A table gives: at $x=1$, $f=3,f^{\prime }=-2,g=4,g^{\prime }=2$; at $x=2$, $f=1,f^{\prime }=4,g=5,g^{\prime }=3$; at $x=3$, $f=2,f^{\prime }=1,g=7,g^{\prime }=5$. If $k(x)=g(f(x))$, find $k^{\prime }(1)$.

A table gives: at $x=-1$, $f=4,f^{\prime }=3,g=2,g^{\prime }=1$; at $x=0$, $f=2,f^{\prime }=-1,g=-1,g^{\prime }=4$; at $x=1$, $f=5,f^{\prime }=2,g=3,g^{\prime }=-2$. If $k(x)=f(g(x))$, find $k^{\prime }(0)$.

A table gives: at $x=0$, $f=4,f^{\prime }=1,g=3,g^{\prime }=-4$; at $x=2$, $f=0,f^{\prime }=5,g=6,g^{\prime }=2$; at $x=3$, $f=2,f^{\prime }=1,g=1,g^{\prime }=3$. If $k(x)=g(f(x))$, find $k^{\prime }(2)$.

The graph shows linear functions $f$ and $g$. For $h(x)=f(x)g(x)$, use the graph to find $h^{\prime }(2)$.

The graph shows linear functions $f$ (the labeled line) and $g$ (the unlabeled line). For $q(x)=\frac{f(x)}{g(x)}$, use the graph to find $q^{\prime }(1)$.

The graph shows linear functions $f$ and $g$, with $f$ labeled. For $q(x)=\frac{f(x)}{g(x)}$, use the graph to find $q^{\prime }(3)$.

The graph shows linear functions $f$ and $g$. For $h(x)=f(x)g(x)$, use the graph to find $h^{\prime }(2)$.

The graph shows functions $f$ and $g$, with $f$ labeled. For $k(x)=f(g(x))$, use the graph to find $k^{\prime }(2)$.

The volume of liquid in a tank is $V(t)=30t-t^2$ liters, where $t$ is minutes. Find $V^{\prime }(4)$ and its meaning.

A cart's position is $s(t)=t^3-6t^2+9t$ meters, where $t$ is seconds. What are $s^{\prime }(2)$ and its meaning?

A greenhouse temperature is $T(t)=18+4t-0.5t^2$ degrees Celsius, where $t$ is hours. Find $T^{\prime }(3)$ and interpret it.

The mass of a dissolving sample is $M(t)=\frac{100}{t+1}$ grams, where $t$ is minutes. Find $M^{\prime }(1)$ and interpret it.

A circular spill has radius $r(t)=2t+1$ meters, where $t$ is minutes, and area $A=\pi r^2$. Find $A^{\prime }(1)$.

At $x=1$, select the correct unsimplified expression for the derivative of $y=(3x^2+1{)}^5$.

For the curve $p(x)=x^2\sin x$, select the correct unsimplified expression for $p^{\prime }\left(\frac{\pi }{2}\right)$.

The curve $x^2+xy+y^2=7$ passes through $(1,2)$. Select the correct unsimplified expression for $\frac{dy}{dx}$ at that point.

Function $g$ is the inverse of differentiable function $f$. Given $f(2)=5$ and $f^{\prime }(2)=-3$, select the expression that gives $g^{\prime }(5)$.

A student writes: “If $h(x)=f(x)g(x)$, $f^{\prime }(2)=0$, and $f(2)=0$, then $h^{\prime }(2)=0$.” Is the reasoning valid?

A student writes: “Because $k(x)=f(g(x))$ and $f^{\prime }(g(1))=6$, it follows that $k^{\prime }(1)=6$.” Is the reasoning valid?

A student differentiates $x^2+y^2=25$ as $2x+2y=0$ and concludes that at $(3,4)$, $\frac{dy}{dx}=-\frac{3}{4}$. Is the reasoning valid?