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Mathematics LibreTexts

13.5E: The Chain Rule for Functions of Multiple Variables (Exercises)

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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.

Frustum of a cone with height z, top radius of x and bottom radius of y.

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})

 


This page titled 13.5E: The Chain Rule for Functions of Multiple Variables (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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