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


13.5: The Chain Rule

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

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

Answer:
( dfrac{dw}{dt}=ycos z+xcos z(2t)−dfrac{xysin 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=5cos(2t),) and ( y=5sin(2t)), find ( dfrac{∂w}{∂t}).

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

Answer:
( dfrac{∂f}{∂r}=rsin(2θ))

6) Suppose ( f(x,y)=x+y,u=e^xsin y,quad x=t^2) and ( y=πt), where ( x=rcos θ) and ( y=rsin 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^xsin 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}ig(pi - 4ig))

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

16) ( sin(6x)+ an(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) ( xcos(xy)+ycos 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=3cos 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=2cos u,) and ( y=3sin v.) Find ( dfrac{∂z}{∂u}) and ( dfrac{∂z}{∂v}).

Answer:
( dfrac{∂z}{∂u}=dfrac{−2sin u}{3sin v}) and ( dfrac{∂z}{∂v}=dfrac{−2cos ucos v}{3sin^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=rcos θ,) and ( y=rsin 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}=(2−4sqrt{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}=-3yzcos(xyz)−xze^{1−t}cos(xyz)+4xycos(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: ( xdfrac{∂f}{∂x}+ydfrac{∂f}{∂y}=nf(x,y)). Show that the given function is homogeneous and verify that ( xdfrac{∂f}{∂x}+ydfrac{∂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}=xfrac{1}{2}(x^2+y^2)^{−1/2}×2x+yfrac{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:
( dfrac{dV}{dt} = frac{34π}{3}, ext{units}^3/ ext{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:
( frac{dV}{dt}=frac{1066π}{3}, ext{cm}^3/ ext{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:
( frac{dA}{dt}=12, ext{in.}^2/ ext{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/ ext{min},) the ( y) resistance is changing at the rate of ( 1,Ω/ ext{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}))

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Paul Seeburger (Monroe Community College) edited the LaTeX.