The Chain Rule for Functions of Multiple Variables (Exercises)
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- Jun 14, 2019
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( \newcommand{\kernel}{\mathrm{null}\,}\)
In exercises 1 - 6, use the information provided to solve the problem.
1) Let w(x,y,z)=xycosz, where x=t,y=t2, and z=arcsint. Find dwdt.
- Answer
- dwdt=ycosz+xcosz(2t)−xysinz√1−t2
2) Let w(t,v)=etv where t=r+s and v=rs. Find ∂w∂r and ∂w∂s.
3) If w=5x2+2y2,x=−3u+v, and y=u−4v, find ∂w∂u and ∂w∂v.
- Answer
- ∂w∂u=−30x+4y =−30(−3u+v)+4(u−4v)=90u−30v+4u−16v=94u−46v,
∂w∂v=10x−16y =10(−3u+v)−16(u−4v)=−30u+10v−16u+64v=−46u+74v
4) If w=xy2,x=5cos(2t), and y=5sin(2t), find ∂w∂t.
5) If f(x,y)=xy,x=rcosθ, and y=rsinθ, find ∂f∂r and express the answer in terms of r and θ.
- Answer
- ∂f∂r=rsin(2θ)
6) Suppose f(x,y)=x+y,u=exsiny,x=t2 and y=πt, where x=rcosθ and y=rsinθ. Find ∂f∂θ.
In exercises 7 - 12, find dzdt in two ways, first using the chain rule and then by direct substitution.
7) z=x2+y2,x=t,y=t2
- Answer
- dzdt=2t+4t3
8) z=√x2+y2,y=t2,x=t
9) z=xy,x=1−√t,y=1+√t
- Answer
- dzdt=−1
10) z=xy,x=et,y=2et
11) z=ln(x+y),x=et,y=et
- Answer
- dzdt=1
12) z=x4,x=t,y=t
13) Let w(x,y,z)=x2+y2+z2,x=cost,y=sint, and z=et. Express w as a function of t and find dwdt directly. Then, find dwdt using the chain rule.
- Answer
- dwdt=2e2t in both cases
14) Let z=x2y, where x=t2 and y=t3. Find dzdt.
15) Let u=exsiny, where x=−ln2t and y=πt. Find dudt when x=ln2 and y=π4.
- Answer
- dudt=√2(π−4)
In exercises 16 - 33, find dydx using partial derivatives.
16) sin(6x)+tan(8y)+5=0
17) x3+y2x−3=0
- Answer
- dydx=−3x2+y22xy
18) sin(x+y)+cos(x−y)=4
19) x2−2xy+y4=4
- Answer
- dydx=y−x−x+2y3
20) xey+yex−2x2y=0
21) x2/3+y2/3=a2/3
- Answer
- dydx=−3√yx
22) xcos(xy)+ycosx=2
23) exy+yey=1
- Answer
- dydx=−yexyxexy+ey(1+y)
24) x2y3+cosy=0
25) Find dzdt using the chain rule where z=3x2y3,x=t4, and y=t2.
- Answer
- dzdt=42t13
26) Let z=3cosx−sin(xy),x=1t, and y=3t. Find dzdt.
27) Let z=e1−xy,x=t1/3, and y=t3. Find dzdt.
- Answer
- dzdt=−103t7/3×e1−t10/3
28) Find dzdt by the chain rule where z=cosh2(xy),x=12t, and y=et.
29) Let z=xy,x=2cosu, and y=3sinv. Find ∂z∂u and ∂z∂v.
- Answer
- ∂z∂u=−2sinu3sinv and ∂z∂v=−2cosucosv3sin2v
30) Let z=ex2y, where x=√uv and y=1v. Find ∂z∂u and ∂z∂v.
31) If z=xyex/y,x=rcosθ, and y=rsinθ, find ∂z∂r and ∂z∂θ when r=2 and θ=π6.
- Answer
- ∂z∂r=√3e√3,∂z∂θ=(2−4√3)e√3
32) Find ∂w∂s if w=4x+y2+z3,x=ers2,y=ln(r+st), and z=rst2.
33) If w=sin(xyz),x=1−3t,y=e1−t, and z=4t, find ∂w∂t.
- Answer
- ∂w∂t=−3yzcos(xyz)−xze1−tcos(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)=tnf(x,y). For all homogeneous functions of degree n, the following equation is true: x∂f∂x+y∂f∂y=nf(x,y). Show that the given function is homogeneous and verify that x∂f∂x+y∂f∂y=nf(x,y).
34) f(x,y)=3x2+y2
35) f(x,y)=√x2+y2
- Answer
- f(tx,ty)=√t2x2+t2y2=t1f(x,y),∂f∂y=x12(x2+y2)−1/2×2x+y12(x2+y2)−1/2×2y=1f(x,y)
36) f(x,y)=x2y−2y3
37) The volume of a right circular cylinder is given by V(x,y)=πx2y, 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=12t and y=13t 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
- \ddfracdVdt=34π3units3/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 dPdt when k=1,dVdt=2 cm3/min, dTdt=12 K/min, V=20cm3, 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})