### Chain Rule

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1. Use the chain rule to find $\frac{dw}{dt}$ if $w=x{e}^{y/z}$, $x={t}^{2}$, $y=1-t$, and $z=1+2t$.
${e}^{y/z}\left(2t-\frac{x}{z}-\frac{2xy}{{z}^{2}}\right)$
2. Use the chain rule to find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ if $z={x}^{2}+xy+{y}^{2}$, $x=s+t$, and $y=st$.
$\frac{\partial z}{\partial s}=2x+y+xt+2yt$,
$\frac{\partial z}{\partial t}=2x+y+xs+2ys$.
3. Use the chain rule to find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ if $z=\mathrm{sin}\left(\alpha \right)\mathrm{tan}\left(\beta \right)$, $\alpha =3s+t$, and $\beta =s-t$.
4. If $z=f\left(x,y\right)$ where $f$ is differentiable, $x=g\left(t\right)$, $y=h\left(t\right)$, $g\left(3\right)=2$, $g\prime \left(3\right)=5$, $h\left(3\right)=7$, $h\prime \left(3\right)=-4$, ${f}_{x}\left(2,7\right)=6$, and ${f}_{y}\left(2,7\right)=-8$, find $\frac{dz}{dt}$ when $t=3$.
$62$
5. Let $W\left(s,t\right)=F\left(u\left(s,t\right),v\left(s,t\right)\right)$, where $F$, $u$, and $v$ are differentiable, $u\left(1,0\right)=2$, ${u}_{s}\left(1,0\right)=-2$, ${u}_{t}\left(1,0\right)=6$, $v\left(1,0\right)=3$, ${v}_{s}\left(1,0\right)=5$, ${v}_{t}\left(1,0\right)=4$, ${F}_{u}\left(2,3\right)=-1$, ${F}_{v}\left(2,3\right)=10$. Find ${W}_{s}\left(1,0\right)$ and ${W}_{t}\left(1,0\right)$.
1. Suppose that $f$ is a differentiable function of $x$ and $y$ and

$g\left(u,v\right)=f\left({e}^{u}+\mathrm{sin}\left(v\right),{e}^{u}+\mathrm{cos}\left(v\right)\right)$.

Use the table of values to calculate ${g}_{u}\left(0,0\right)$ and ${g}_{v}\left(0,0\right)$.

2. Suppose that $f$ is a differentiable function of $x$ and $y$ and

$g\left(r,s\right)=f\left(2r-s,{s}^{2}-4r\right)$.

Use the table of values to calculate ${g}_{r}\left(1,2\right)$ and ${g}_{s}\left(1,2\right)$.

$f$$g$${f}_{x}$${f}_{y}$
$\left(0,0\right)$$3$$6$$4$$8$
$\left(1,2\right)$$6$$3$$2$$5$

1. $7$ and $2$.
6. Use the tree diagram to write out the chain rule for differentiating $u=f\left(x,y\right)$, where $x=x\left(r,s,t\right)$ and $y=y\left(r,s,t\right)$.
$\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}$,
$\frac{\partial u}{\partial s}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial s}$,
$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial t}$.
7. Let $z={x}^{2}+x{y}^{3}$, $x=u{v}^{2}+{w}^{3}$, and $y=u+v{e}^{w}$. Use the chain rule to find $\frac{\partial z}{\partial u}$, $\frac{\partial z}{\partial v}$, and $\frac{\partial z}{\partial w}$ at $\left(u,v,w\right)=\left(2,1,0\right)$.
$85$, $178$, and $54$.
8. The temperature at a point $\left(x,y\right)$ is $T\left(x,y\right)$, measured in kelvin. A Mars Exploration Rover crawls so that its position after $t$ seconds is given by $x=\sqrt{1+t}$ and $y=2+\frac{1}{3}t$, where $x$ and $y$ are in centimeters. The temperature function satisfies ${T}_{x}\left(2,3\right)=4$ and ${T}_{y}\left(2,3\right)=3$. How fast is the temperature rising on the rover's path after $3$ seconds?
$2$ [K/s].
9. Soylent Green production in a given year, $W$, depends on the average temperature $T$ and the annual rainfall $R$. It is estimated that the average temperature is rising at a rate of $0.15$ K/year and rainfall is decreasing at a rate of $0.1$ cm/year. It is also known that at current production levels, $\frac{\partial W}{\partial T}=-2$ and $\frac{\partial W}{\partial R}=8$. What is the significance of the signs of these partial derivatives? Estimate the current rate of change of Soylent Green production, $\frac{dW}{dt}$.
10. The radius of a right circular cone is increasing at a rate of $1.8$ cm/s and its height is decreasing at the rate of $2.5$ cm/s. At what rate is the volume of the cone changing when the radius is $120$ cm and the height is $140$ cm?
Approximately $25635$ cubic cm per second.
11. The length $l$, width $w$, and height $h$ of a rectangular box change with respect to time $t$. At a certain time the dimensions are $\left(l,w,h\right)=\left(1,2,2\right)$. At the same time, $l$ and $w$ are increasing at a rate of $2$ m/s, while $h$ is decreasing at a rate of $3$ m/s. Find the rates at which the following are changing:
1. The volume of the box.
2. The surface area of the box.
3. The length of the diagonal of the box.
1. $6$ cubic meters per second.
2. $10$ square meters per second.
3. $0$ m/s.
12. Assuming that $f$ is differentiable and $z=f\left(x-y\right)$, show that $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0$.