The Chain Rule for Functions of Multiple Variables (Exercises)

In exercises 1 - 6, use the information provided to solve the problem.

1) Let $w(x,y,z)=xy\cos z,$ where $x=t,y=t^2,$ and $z=\arcsin t.$ Find $\dfrac{dw}{dt}$.

Answer

$\dfrac{dw}{dt}=y\cos z+x\cos z(2t)−\dfrac{xy\sin z}{\sqrt{1−t^2}}$

2) Let $w(t,v)=e^{tv}$ where $t=r+s$ and $v=rs$. Find $\dfrac{∂w}{∂r}$ and $\dfrac{∂w}{∂s}$.

3) If $w=5x^2+2y^2, \quad x=−3u+v,$ and $y=u−4v,$ find $\dfrac{∂w}{∂u}$ and $\dfrac{∂w}{∂v}$.

Answer

$\dfrac{∂w}{∂u}=−30x+4y \quad\ = \quad -30(-3u + v) + 4(u - 4v) \quad = \quad 90u -30v + 4u - 16v \quad = \quad 94u - 46v$,
$\dfrac{∂w}{∂v}=10x−16y \quad\ = \quad 10(-3u + v) - 16(u - 4v) \quad = \quad -30u +10v - 16u + 64v \quad = \quad -46u + 74v$

4) If $w=xy^2,x=5\cos(2t),$ and $y=5\sin(2t)$, find $\dfrac{∂w}{∂t}$.

5) If $f(x,y)=xy,x=r\cos θ,$ and $y=r\sin θ$, find $\dfrac{∂f}{∂r}$ and express the answer in terms of $r$ and $θ$.

Answer

$\dfrac{∂f}{∂r}=r\sin(2θ)$

6) Suppose $f(x,y)=x+y,u=e^x\sin y,\quad x=t^2$ and $y=πt$, where $x=r\cos θ$ and $y=r\sin θ$. Find $\dfrac{∂f}{∂θ}$.

In exercises 7 - 12, find $\dfrac{dz}{dt}$ in two ways, first using the chain rule and then by direct substitution.

7) $z=x^2+y^2, \quad x=t,y=t^2$

Answer

$\dfrac{dz}{dt}=2t+4t^3$

8) $z=\sqrt{x^2+y^2},\quad y=t^2,x=t$

9) $z=xy,\quad x=1−\sqrt{t},y=1+\sqrt{t}$

Answer

$\dfrac{dz}{dt}=−1$

10) $z=\frac{x}{y},\quad x=e^t,y=2e^t$

11) $z=\ln(x+y), \quad x=e^t,y=e^t$

Answer

$\dfrac{dz}{dt}=1$

12) $z=x^4,\quad x=t,y=t$

13) Let $w(x,y,z)=x^2+y^2+z^2, \quad x=cost,y=sint,$ and $z=e^t$. Express $w$ as a function of $t$ and find $\dfrac{dw}{dt}$ directly. Then, find $\dfrac{dw}{dt}$ using the chain rule.

Answer

$\dfrac{dw}{dt}=2e^{2t}$ in both cases

14) Let $z=x^2y,$ where $x=t^2$ and $y=t^3$. Find $\dfrac{dz}{dt}$.

15) Let $u=e^x\sin y,$ where $x=-\ln 2t$ and $y=πt$. Find $\dfrac{du}{dt}$ when $x=\ln 2$ and $y=\frac{π}{4}$.

Answer

$\dfrac{du}{dt} = \sqrt{2}\big(\pi - 4\big)$

In exercises 16 - 33, find $\dfrac{dy}{dx}$ using partial derivatives.

16) $\sin(6x)+\tan(8y)+5=0$

17) $x^3+y^2x−3=0$

Answer

$\dfrac{dy}{dx}=−\dfrac{3x^2+y^2}{2xy}$

18) $\sin(x+y)+\cos(x−y)=4$

19) $x^2−2xy+y^4=4$

Answer

$\dfrac{dy}{dx}=\dfrac{y−x}{−x+2y^3}$

20) $xe^y+ye^x−2x^2y=0$

21) $x^{2/3}+y^{2/3}=a^{2/3}$

Answer

$\dfrac{dy}{dx}=−\sqrt[3]{\frac{y}{x}}$

22) $x\cos(xy)+y\cos x=2$

23) $e^{xy}+ye^y=1$

Answer

$\dfrac{dy}{dx}=−\dfrac{ye^{xy}}{xe^{xy}+e^y(1+y)}$

24) $x^2y^3+\cos y=0$

25) Find $\dfrac{dz}{dt}$ using the chain rule where $z=3x^2y^3,\,\,x=t^4,$ and $y=t^2$.

Answer

$\dfrac{dz}{dt}=42t^{13}$

26) Let $z=3\cos x−\sin(xy),x=\frac{1}{t},$ and $y=3t.$ Find $\dfrac{dz}{dt}$.

27) Let $z=e^{1−xy},\,\, x=t^{1/3},$ and $y=t^3$. Find $\dfrac{dz}{dt}$.

Answer

$\dfrac{dz}{dt}=−\frac{10}{3}t^{7/3}×e^{1−t^{10/3}}$

28) Find $\dfrac{dz}{dt}$ by the chain rule where $z=\cosh^2(xy),\,\,x=\frac{1}{2}t,$ and $y=e^t$.

29) Let $z=\dfrac{x}{y},\,\, x=2\cos u,$ and $y=3\sin v.$ Find $\dfrac{∂z}{∂u}$ and $\dfrac{∂z}{∂v}$.

Answer

$\dfrac{∂z}{∂u}=\dfrac{−2\sin u}{3\sin v}$ and $\dfrac{∂z}{∂v}=\dfrac{−2\cos u\cos v}{3\sin^2v}$

30) Let $z=e^{x^2y}$, where $x=\sqrt{uv}$ and $y=\frac{1}{v}$. Find $\dfrac{∂z}{∂u}$ and $\dfrac{∂z}{∂v}$.

31) If $z=xye^{x/y},\,\, x=r\cos θ,$ and $y=r\sin θ$, find $\dfrac{∂z}{∂r}$ and $\dfrac{∂z}{∂θ}$ when $r=2$ and $θ=\frac{π}{6}$.

Answer

$\dfrac{∂z}{∂r}=\sqrt{3}e^{\sqrt{3}}, \dfrac{∂z}{∂θ}=(2−4\sqrt{3})e^{\sqrt{3}}$

32) Find $\dfrac{∂w}{∂s}$ if $w=4x+y^2+z^3,\,\,x=e^{rs^2},\,\,y=\ln(\frac{r+s}{t}),$ and $z=rst^2$.

33) If $w=\sin(xyz),\,\,x=1−3t,\,\,y=e^{1−t},$ and $z=4t$, find $\dfrac{∂w}{∂t}$.

Answer

$\dfrac{∂w}{∂t}=-3yz\cos(xyz)−xze^{1−t}\cos(xyz)+4xy\cos(xyz)$

In exercises 34 - 36, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)=t^nf(x,y)$. For all homogeneous functions of degree $n$, the following equation is true: $x\dfrac{∂f}{∂x}+y\dfrac{∂f}{∂y}=nf(x,y)$. Show that the given function is homogeneous and verify that $x\dfrac{∂f}{∂x}+y\dfrac{∂f}{∂y}=nf(x,y)$.

34) $f(x,y)=3x^2+y^2$

35) $f(x,y)=\sqrt{x^2+y^2}$

Answer

$f(tx,ty)=\sqrt{t^2x^2+t^2y^2}=t^1f(x,y), \quad \dfrac{∂f}{∂y}=x\frac{1}{2}(x^2+y^2)^{−1/2}×2x+y\frac{1}{2}(x^2+y^2)^{−1/2}×2y=1f(x,y)$

36) $f(x,y)=x^2y−2y^3$

37) The volume of a right circular cylinder is given by $V(x,y)=πx^2y,$ where $x$ is the radius of the cylinder and $y$ is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x=\frac{1}{2}t$ and $y=\frac{1}{3}t$ so that $x$ and $y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=5$? Assume time is measured in seconds.

Answer

$\ddfrac{dV}{dt} = \frac{34π}{3}\,\text{units}^3/\text{s}$

38) The pressure $P$ of a gas is related to the volume and temperature by the formula $PV=kT$, where temperature is expressed in kelvins. Express the pressure of the gas as a function of both $V$ and $T$. Find $\dfrac{dP}{dt}$ when $k=1, \dfrac{dV}{dt}=2$ cm³/min, $\dfrac{dT}{dt}=12$ K/min, $V=20 cm^3$, and $T=20°F$.

39) The radius of a right circular cone is increasing at $3$ cm/min whereas the height of the cone is decreasing at $2$ cm/min. Find the rate of change of the volume of the cone when the radius is $13$ cm and the height is $18$ cm.

Answer

$\dfrac{dV}{dt}=\frac{1066π}{3}\,\text{cm}^3/\text{min}$

40) The volume of a frustum of a cone is given by the formula $V=\frac{1}{3}πz(x^2+y^2+xy),$ where $x$ is the radius of the smaller circle, $y$ is the radius of the larger circle, and $z$ is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when $x=10$ in., $y=12$ in., and $z=18$ in.

41) A closed box is in the shape of a rectangular solid with dimensions $x,y,$ and $z$. (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.5$ in./min. Find the rate of change of the total surface area of the box when $x=2$ in., $y=3$ in., and $z=1$ in.

Answer

$\dfrac{dA}{dt}=12\, \text{in.}^2/\text{min}$

42) The total resistance in a circuit that has three individual resistances represented by $x,y,$ and $z$ is given by the formula $R(x,y,z)=\dfrac{xyz}{yz+xz+xy}$. Suppose at a given time the $x$ resistance is $100\,Ω$, the $y$ resistance is $200\,Ω,$ and the $z$ resistance is $300\,Ω.$ Also, suppose the $x$ resistance is changing at a rate of $2\,Ω/\text{min},$ the $y$ resistance is changing at the rate of $1\,Ω/\text{min}$, and the $z$ resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

43) The temperature $T$ at a point $(x,y)$ is $T(x,y)$ and is measured using the Celsius scale. A fly 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 measured in centimeters. The temperature function satisfies $T_x(2,3)=4$ and $T_y(2,3)=3$. How fast is the temperature increasing on the fly’s path after $3$ sec?

Answer

$2$°C/sec

44) The $x$ and $y$ components of a fluid moving in two dimensions are given by the following functions: $u(x,y)=2y$ and $v(x,y)=−2x$ with $x≥0$ and $y≥0$. The speed of the fluid at the point $(x,y)$ is $s(x,y)=\sqrt{u(x,y)^2+v(x,y)^2}$. Find $\dfrac{∂s}{∂x}$ and $\dfrac{∂s}{∂y}$ using the chain rule.

45) Let $u=u(x,y,z),$ where $x=x(w,t),\, y=y(w,t),\, z=z(w,t),\, w=w(r,s)$, and $t=t(r,s).$ Use a tree diagram and the chain rule to find an expression for $\dfrac{∂u}{∂r}$.

Answer

$\frac{∂u}{∂r}=\frac{∂u}{∂x}(\frac{∂x}{∂w}\frac{∂w}{∂r}+\frac{∂x}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂y}(\frac{∂y}{∂w}\frac{∂w}{∂r}+\frac{∂y}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂z}(\frac{∂z}{∂w}\frac{∂w}{∂r}+\frac{∂z}{∂t}\frac{∂t}{∂r})$