Chain Rule

(requires JavaScript)

1. Use the chain rule to find ${\displaystyle \frac{dw}{dt}}$ if $w=x{e}^{y/z}$, $x=t^2$, $y=1-t$, and $z=1+2t$.
   ${\displaystyle e^{y/z}\left(2t-\frac{x}{z}-\frac{2xy}{z^2}\right)}$
2. Use the chain rule to find ${\displaystyle \frac{\partial z}{\partial s}}$ and ${\displaystyle \frac{\partial z}{\partial t}}$ if $z=x^2+xy+y^2$, $x=s+t$, and $y=st$.
   ${\displaystyle \frac{\partial z}{\partial s}}=2x+y+xt+2yt$,
   ${\displaystyle \frac{\partial z}{\partial t}}=2x+y+xs+2ys$.
3. Use the chain rule to find ${\displaystyle \frac{\partial z}{\partial s}}$ and ${\displaystyle \frac{\partial z}{\partial t}}$ if $z=\sin <!--FUNCTION\; APPLICATION-->\left(\alpha \right)\tan <!--FUNCTION\; APPLICATION-->\left(\beta \right)$, $\alpha =3s+t$, and $\beta =s-t$.
4. If $z=f<!--FUNCTION\; APPLICATION-->\left(x,y\right)$ where $f$ is differentiable, $x=g<!--FUNCTION\; APPLICATION-->\left(t\right)$, $y=h<!--FUNCTION\; APPLICATION-->\left(t\right)$, $g<!--FUNCTION\; APPLICATION-->\left(3\right)=2$, $g\prime <!--FUNCTION\; APPLICATION-->\left(3\right)=5$, $h<!--FUNCTION\; APPLICATION-->\left(3\right)=7$, $h\prime <!--FUNCTION\; APPLICATION-->\left(3\right)=-4$, $f_x<!--FUNCTION\; APPLICATION-->\left(2,7\right)=6$, and $f_y<!--FUNCTION\; APPLICATION-->\left(2,7\right)=-8$, find ${\displaystyle \frac{dz}{dt}}$ when $t=3$.
   $62$
5. Let $W<!--FUNCTION\; APPLICATION-->\left(s,t\right)=F<!--FUNCTION\; APPLICATION-->\left(u<!--FUNCTION\; APPLICATION-->\left(s,t\right),v<!--FUNCTION\; APPLICATION-->\left(s,t\right)\right)$, where $F$, $u$, and $v$ are differentiable, $u<!--FUNCTION\; APPLICATION-->\left(1,0\right)=2$, $u_s<!--FUNCTION\; APPLICATION-->\left(1,0\right)=-2$, $u_t<!--FUNCTION\; APPLICATION-->\left(1,0\right)=6$, $v<!--FUNCTION\; APPLICATION-->\left(1,0\right)=3$, $v_s<!--FUNCTION\; APPLICATION-->\left(1,0\right)=5$, $v_t<!--FUNCTION\; APPLICATION-->\left(1,0\right)=4$, $F_u<!--FUNCTION\; APPLICATION-->\left(2,3\right)=-1$, $F_v<!--FUNCTION\; APPLICATION-->\left(2,3\right)=10$. Find $W_s<!--FUNCTION\; APPLICATION-->\left(1,0\right)$ and $W_t<!--FUNCTION\; APPLICATION-->\left(1,0\right)$.

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

      $g<!--FUNCTION\; APPLICATION-->\left(u,v\right)=f<!--FUNCTION\; APPLICATION-->\left(e^u+\sin <!--FUNCTION\; APPLICATION-->\left(v\right),e^u+\cos <!--FUNCTION\; APPLICATION-->\left(v\right)\right)$.

      Use the table of values to calculate $g_u<!--FUNCTION\; APPLICATION-->\left(0,0\right)$ and $g_v<!--FUNCTION\; APPLICATION-->\left(0,0\right)$.

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

      $g<!--FUNCTION\; APPLICATION-->\left(r,s\right)=f<!--FUNCTION\; APPLICATION-->\left(2r-s,s^2-4r\right)$.

      Use the table of values to calculate $g_r<!--FUNCTION\; APPLICATION-->\left(1,2\right)$ and $g_s<!--FUNCTION\; APPLICATION-->\left(1,2\right)$.

   	$f$	$g$	$f_x$	$f_y$
   $\left(0,0\right)$	$3$	$6$	$4$	$8$
   $\left(1,2\right)$	$6$	$3$	$2$	$5$

   1. $7$ and $2$.
7. Use the tree diagram to write out the chain rule for differentiating $u=f<!--FUNCTION\; APPLICATION-->\left(x,y\right)$, where $x=x<!--FUNCTION\; APPLICATION-->\left(r,s,t\right)$ and $y=y<!--FUNCTION\; APPLICATION-->\left(r,s,t\right)$.
   ${\displaystyle \frac{\partial u}{\partial r}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial r}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial r}}$,
   ${\displaystyle \frac{\partial u}{\partial s}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial s}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial s}}$,
   ${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial x}{\partial t}}+{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{\partial y}{\partial t}}$.
8. 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 ${\displaystyle \frac{\partial z}{\partial u}}$, ${\displaystyle \frac{\partial z}{\partial v}}$, and ${\displaystyle \frac{\partial z}{\partial w}}$ at $\left(u,v,w\right)=\left(2,1,0\right)$.
   $85$, $178$, and $54$.
9. The temperature at a point $\left(x,y\right)$ is $T<!--FUNCTION\; APPLICATION-->\left(x,y\right)$, 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 $T_x<!--FUNCTION\; APPLICATION-->\left(2,3\right)=4$ and $T_y<!--FUNCTION\; APPLICATION-->\left(2,3\right)=3$. How fast is the temperature rising on the rover's path after $3$ seconds?
   $2$ [K/s].
10. 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, ${\displaystyle \frac{\partial W}{\partial T}}=-2$ and ${\displaystyle \frac{\partial W}{\partial R}}=8$. What is the significance of the signs of these partial derivatives? Estimate the current rate of change of Soylent Green production, ${\displaystyle \frac{dW}{dt}}$.
11. 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.
12. 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.
13. Assuming that $f$ is differentiable and $z=f<!--FUNCTION\; APPLICATION-->\left(x-y\right)$, show that ${\displaystyle \frac{\partial z}{\partial x}}+{\displaystyle \frac{\partial z}{\partial y}}=0$.