
Use the chain rule to find $\frac{dw}{dt}$
if $w=x{e}^{y/z}$,
$x={t}^{2}$,
$y=1t$,
and $z=1+2t$.
${e}^{y/z}\left(2t\frac{x}{z}\frac{2xy}{{z}^{2}}\right)$

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 $y=st$.
$\frac{\partial z}{\partial s}}=2x+y+xt+2yt$,
$\frac{\partial z}{\partial t}}=2x+y+xs+2ys$.

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 $\beta =st$.

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 $t=3$.
$62$

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 ${W}_{t}\left(1,0\right)$.


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 ${g}_{v}\left(0,0\right)$.

Suppose that $f$ is a differentiable function of $x$ and
$y$ and
$g\left(r,s\right)=f\left(2rs,{s}^{2}4r\right)$.
Use the table of values to calculate ${g}_{r}\left(1,2\right)$
and ${g}_{s}\left(1,2\right)$.
 $f$  $g$  ${f}_{x}$  ${f}_{y}$ 
$\left(0,0\right)$  $3$  $6$  $4$  $8$ 
$\left(1,2\right)$  $6$  $3$  $2$  $5$ 

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 $y=y\left(r,s,t\right)$.
$\frac{\partial u}{\partial r}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial r}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial r}$,
$\frac{\partial u}{\partial s}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial s}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial s}$,
$\frac{\partial u}{\partial t}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial t}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial t}$.

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 $54$.

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].

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, $\frac{dW}{dt}$.

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.

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:
 The volume of the box.
 The surface area of the box.
 The length of the diagonal of the box.
 $6$ cubic meters per second.
 $10$ square meters per second.
 $0$ m/s.

Assuming that $f$ is differentiable and $z=f\left(xy\right)$,
show that $\frac{\partial z}{\partial x}}+{\displaystyle \frac{\partial z}{\partial y}}=0$.