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# 6.4E: Exercises

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## Exercise $$\PageIndex{1}$$

For the following exercises, use the information provided to solve the problem.

1) Let $$\displaystyle w(x,y,z)=xycosz,$$ where $$\displaystyle x=t,y=t^2,$$ and $$\displaystyle z=arcsint.$$ Find $$\displaystyle \frac{dw}{dt}$$.

$$\displaystyle \frac{dw}{dt}=ycosz+xcosz(2t)−\frac{xysinz}{\sqrt{1−t^2}}$$

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

3) If $$\displaystyle w=5x^2+2y^2,x=−3s+t,$$ and $$\displaystyle y=s−4t,$$ find $$\displaystyle \frac{∂w}{∂s}$$ and $$\displaystyle \frac{∂w}{∂t}$$.

$$\displaystyle \frac{∂w}{∂s}=−30x+4y, \frac{∂w}{∂t}=10x−16y$$

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

5) If $$\displaystyle f(x,y)=xy,x=rcosθ,$$ and $$\displaystyle y=rsinθ$$, find ∂f∂r and express the answer in terms of $$\displaystyle r$$ and $$\displaystyle θ$$.

$$\displaystyle \frac{∂f}{∂r}=rsin(2θ)$$

6) Suppose $$\displaystyle f(x,y)=x+y,u=e^xsiny,x=t^2$$ and $$\displaystyle y=πt$$, where $$\displaystyle x=rcosθ$$ and $$\displaystyle y=rsinθ$$. Find $$\displaystyle \frac{∂f}{∂θ}$$.

## Exercise $$\PageIndex{2}$$

For the following exercises, find $$\displaystyle \frac{df}{dt}$$ using the chain rule and direct substitution.

7) $$\displaystyle f(x,y)=x^2+y^2, x=t,y=t^2$$

$$\displaystyle \frac{df}{dt}=2t+4t^3$$

8) $$\displaystyle f(x,y)=\sqrt{x^2+y^2},y=t^2,x=t$$

9) $$\displaystyle f(x,y)=xy,x=1−\sqrt{t},y=1+\sqrt{t}$$

$$\displaystyle \frac{df}{dt}=−1$$

10) $$\displaystyle f(x,y)=\frac{x}{y},x=e^t,y=2e^t$$

11) $$\displaystyle f(x,y)=ln(x+y), x=e^t,y=e^t$$

$$\displaystyle \frac{df}{dt}=1$$

12) $$\displaystyle f(x,y)=x^4, x=t,y=t$$

## Exercise $$\PageIndex{3}$$

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

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

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

15) Let $$\displaystyle u=e^xsiny,$$ where $$\displaystyle x=t^2$$ and $$\displaystyle y=πt$$. Find $$\displaystyle \frac{du}{dt}$$ when $$\displaystyle x=ln2$$ and $$\displaystyle y=\frac{π}{4}$$.

$$\displaystyle 2\sqrt{2}t+\sqrt{2}π=\frac{du}{dt}$$

## Exercise $$\PageIndex{4}$$

For the following exercises, find $$\displaystyle \frac{dy}{dx}$$using partial derivatives.

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

17) $$\displaystyle x^3+y^2x−3=0$$

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

18) $$\displaystyle sin(x+y)+cos(x−y)=4$$

19) $$\displaystyle x^2−2xy+y^4=4$$

$$\displaystyle \frac{dy}{dx}=\frac{y−x}{−x+2y^3}$$

20) $$\displaystyle xe^y+ye^x−2x^2y=0$$

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

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

22) $$\displaystyle xcos(xy)+ycosx=2$$

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

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

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

## Exercise $$\PageIndex{5}$$

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

$$\displaystyle \frac{dz}{dt}=42t^{13}$$

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

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

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

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

29) Let $$\displaystyle z=\frac{x}{y},x=2cosu,$$ and $$\displaystyle y=3sinv.$$ Find $$\displaystyle \frac{∂z}{∂u}$$ and $$\displaystyle \frac{∂z}{∂v}$$.

$$\displaystyle \frac{∂z}{∂u}=\frac{−2sinu3}{sinv}$$ and $$\displaystyle \frac{∂z}{∂v}=\frac{−2cosucosv3}{sin^2v}$$

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

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

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

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

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

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

## Exercise $$\PageIndex{6}$$

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

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

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

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

36) $$\displaystyle f(x,y)=x^2y−2y^3$$

## Exercise $$\PageIndex{7}$$

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

$$\displaystyle \frac{34π}{3}$$

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

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

$$\displaystyle \frac{dV}{dt}=\frac{1066π}{3}cm^3/min$$

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

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

$$\displaystyle \frac{dA}{dt}=12in.^2/min$$

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

43) The temperature $$\displaystyle T$$ at a point $$\displaystyle (x,y)$$ is $$\displaystyle T(x,y)$$ and is measured using the Celsius scale. A fly crawls so that its position after $$\displaystyle t$$ seconds is given by $$\displaystyle x=\sqrt{1+t}$$ and $$\displaystyle y=2+\frac{1}{3}t$$, where $$\displaystyle x$$ and $$\displaystyle y$$ are measured in centimeters. The temperature function satisfies $$\displaystyle T_x(2,3)=4$$ and $$\displaystyle T_y(2,3)=3$$. How fast is the temperature increasing on the fly’s path after $$\displaystyle 3$$ sec?

$$\displaystyle 2°C/sec$$
44) The $$\displaystyle x$$ and $$\displaystyle y$$ components of a fluid moving in two dimensions are given by the following functions: $$\displaystyle u(x,y)=2y$$ and $$\displaystyle v(x,y)=−2x; x≥0;y≥0.$$ The speed of the fluid at the point $$\displaystyle (x,y)$$ is $$\displaystyle s(x,y)=\sqrt{u(x,y)^2+v(x,y)^2}$$. Find $$\displaystyle \frac{∂s}{∂x}$$ and $$\displaystyle \frac{∂s}{∂y}$$ using the chain rule.
45) Let $$\displaystyle u=u(x,y,z),$$ where $$\displaystyle x=x(w,t),y=y(w,t),z=z(w,t),w=w(r,s),$$ and $$\displaystyle t=t(r,s).$$ Use a tree diagram and the chain rule to find an expression for $$\displaystyle \frac{∂u}{∂r}$$.
$$\displaystyle \frac{∂u}{∂r}=\frac{∂u}{∂x}(\frac{∂x}{∂w}\frac{∂w}{∂r}+\frac{∂x}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂y}(\frac{∂y}{∂w}\frac{∂w}{∂r}+\frac{∂y}{∂t}\frac{∂t}{∂r})+\frac{∂u}{∂z}(\frac{∂z}{∂w}\frac{∂w}{∂r}+\frac{∂z}{∂t}\frac{∂t}{∂r})$$