$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.6E: Exercises for Section 3.6

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

In exercises 1 - 6, given $$y=f(u)$$ and $$u=g(x)$$, find $$\dfrac{dy}{dx}$$ by using Leibniz’s notation for the chain rule: $$\dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx}.$$

1) $$y=3u−6,\quad u=2x^2$$

2) $$y=6u^3,\quad u=7x−4$$

$$\dfrac{dy}{dx} = 18u^2⋅7=18(7x−4)^2⋅7= 126(7x−4)^2$$

3) $$y=\sin u,\quad u=5x−1$$

4) $$y=\cos u,\quad u=-\frac{x}{8}$$

$$\dfrac{dy}{dx} = −\sin u⋅\left(-\frac{1}{8}\right)=\frac{1}{8}\sin(-\frac{x}{8})$$

5) $$y=\tan u,\quad u=9x+2$$

6) $$y=\sqrt{4u+3},\quad u=x^2−6x$$

$$\dfrac{dy}{dx} = \dfrac{8x−24}{2\sqrt{4u+3}}=\dfrac{4x−12}{\sqrt{4x^2−24x+3}}$$

For each of the following exercises,

a. decompose each function in the form $$y=f(u)$$ and $$u=g(x)$$, and

b. find $$\dfrac{dy}{dx}$$ as a function of $$x$$.

7) $$y=(3x−2)^6$$

8) $$y=(3x^2+1)^3$$

a. $$f(u)=u^3,\quad u=3x^2+1$$;

b. $$\dfrac{dy}{dx} = 18x(3x^2+1)^2$$

9) $$y=\sin^5(x)$$

10) $$y=\left(\dfrac{x}{7}+\dfrac{7}{x}\right)^7$$

a. $$f(u)=u^7,\quad u=\dfrac{x}{7}+\dfrac{7}{x}$$;

b. $$\dfrac{dy}{dx} = 7\left(\dfrac{x}{7}+\dfrac{7}{x}\right)^6⋅\left(\dfrac{1}{7}−\dfrac{7}{x^2}\right)$$

11) $$y=\tan(\sec x)$$

12) $$y=\csc(πx+1)$$

a. $$f(u)=\csc u,\quad u=πx+1$$;

b. $$\dfrac{dy}{dx} = −π\csc(πx+1)⋅\cot(πx+1)$$

13) $$y=\cot^2x$$

14) $$y=−6\sin^{−3}x$$

a. $$f(u)=−6u^{−3},\quad u=\sin x$$;

b. $$\dfrac{dy}{dx} = 18\sin^{−4}x⋅\cos x$$

In exercises 15 - 24, find $$\dfrac{dy}{dx}$$ for each function.

15) $$y=(3x^2+3x−1)^4$$

16) $$y=(5−2x)^{−2}$$

$$\dfrac{dy}{dx}=\dfrac{4}{(5−2x)^3}$$

17) $$y=\cos^3(πx)$$

18) $$y=(2x^3−x^2+6x+1)^3$$

$$\dfrac{dy}{dx}=6(2x^3−x^2+6x+1)^2⋅(3x^2−x+3)$$

19) $$y=\dfrac{1}{\sin^2(x)}$$

20) $$y=\big(\tan x+\sin x\big)^{−3}$$

$$\dfrac{dy}{dx}=−3\big(\tan x+\sin x\big)^{−4}⋅(\sec^2x+\cos x)$$

21) $$y=x^2\cos^4x$$

22) $$y=\sin(\cos 7x)$$

$$\dfrac{dy}{dx}=−7\cos(\cos 7x)⋅\sin 7x$$

23) $$y=\sqrt{6+\sec πx^2}$$

24) $$y=\cot^3(4x+1)$$

$$\dfrac{dy}{dx}=−12\cot^2(4x+1)⋅\csc^2(4x+1)$$

25) Let $$y=\big[f(x)\big]^3$$ and suppose that $$f′(1)=4$$ and $$\frac{dy}{dx}=10$$ for $$x=1$$. Find $$f(1)$$.

26) Let $$y=\big(f(x)+5x^2\big)^4$$ and suppose that $$f(−1)=−4$$ and $$\frac{dy}{dx}=3$$ when $$x=−1$$. Find $$f′(−1)$$

$$f′(−1)=10\frac{3}{4}$$

27) Let $$y=(f(u)+3x)^2$$ and $$u=x^3−2x$$. If $$f(4)=6$$ and $$\frac{dy}{dx}=18$$ when $$x=2$$, find $$f′(4)$$.

28) [T] Find the equation of the tangent line to $$y=−\sin(\frac{x}{2})$$ at the origin. Use a calculator to graph the function and the tangent line together.

$$y=-\frac{1}{2}x$$

29) [T] Find the equation of the tangent line to $$y=\left(3x+\frac{1}{x}\right)^2$$ at the point $$(1,16)$$. Use a calculator to graph the function and the tangent line together.

30) Find the $$x$$ -coordinates at which the tangent line to $$y=\left(x−\frac{6}{x}\right)^8$$ is horizontal.

$$x=±\sqrt{6}$$

31) [T] Find an equation of the line that is normal to $$g(θ)=\sin^2(πθ)$$ at the point $$\left(\frac{1}{4},\frac{1}{2}\right)$$. Use a calculator to graph the function and the normal line together.

For exercises 32 - 39, use the information in the following table to find $$h′(a)$$ at the given value for $$a$$.

 $$x$$ $$f(x)$$ $$f'(x)$$ $$g(x)$$ $$g'(x)$$ 0 2 5 0 2 1 1 −2 3 0 2 4 4 1 −1 3 3 −3 2 3

32) $$h(x)=f\big(g(x)\big);\quad a=0$$

$$h'(0) = 10$$

33) $$h(x)=g\big(f(x)\big);\quad a=0$$

34) $$h(x)=\big(x^4+g(x)\big)^{−2};\quad a=1$$

$$h'(1) = −\frac{1}{8}$$

35) $$h(x)=\left(\dfrac{f(x)}{g(x)}\right)^2;\quad a=3$$

36) $$h(x)=f\big(x+f(x)\big);\quad a=1$$

$$h'(1) = −4$$

37) $$h(x)=\big(1+g(x)\big)^3;\quad a=2$$

38) $$h(x)=g\big(2+f(x^2)\big);\quad a=1$$

$$h'(1) = −12$$

39) $$h(x)=f\big(g(\sin x)\big);\quad a=0$$

40) [T] The position function of a freight train is given by $$s(t)=100(t+1)^{−2}$$, with $$s$$ in meters and $$t$$ in seconds. At time $$t=6$$ s, find the train’s

a. velocity and

b. acceleration.

c. Considering your results in parts a. and b., is the train speeding up or slowing down?

a. $$v(6) = −\frac{200}{343}$$ m/s,

b. $$a(6) = \frac{600}{2401}\;\text{m/s}^2,$$

c. The train is slowing down since velocity and acceleration have opposite signs.

41) [T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where $$t$$ is measured in seconds and $$s$$ is in inches:

$s(t)=−3\cos\left(πt+\frac{π}{4}\right).\nonumber$

a. Determine the position of the spring at $$t=1.5$$ s.

b. Find the velocity of the spring at $$t=1.5$$ s.

42) [T] The total cost to produce $$x$$ boxes of Thin Mint Girl Scout cookies is $$C$$ dollars, where $$C=0.0001x^3−0.02x^2+3x+300.$$ In $$t$$ weeks production is estimated to be $$x=1600+100t$$ boxes.

a. Find the marginal cost $$C′(x).$$

b. Use Leibniz’s notation for the chain rule, $$\dfrac{dC}{dt}=\dfrac{dC}{dx}⋅\dfrac{dx}{dt}$$, to find the rate with respect to time $$t$$ that the cost is changing.

c. Use your result in part b. to determine how fast costs are increasing when $$t=2$$ weeks. Include units with the answer.

a. $$C′(x)=0.0003x^2−0.04x+3$$

b. $$\dfrac{dC}{dt}=100⋅(0.0003x^2−0.04x+3) = 100⋅(0.0003(1600+100t)^2−0.04(1600+100t)+3) = 300t^2 +9200t +70700$$

c. Approximately \$90,300 per week

43) [T] The formula for the area of a circle is $$A=πr^2$$, where $$r$$ is the radius of the circle. Suppose a circle is expanding, meaning that both the area $$A$$ and the radius $$r$$ (in inches) are expanding.

a. Suppose $$r=2−\dfrac{100}{(t+7)^2}$$ where $$t$$ is time in seconds. Use the chain rule $$\dfrac{dA}{dt}=\dfrac{dA}{dr}⋅\dfrac{dr}{dt}$$ to find the rate at which the area is expanding.

b. Use your result in part a. to find the rate at which the area is expanding at $$t=4$$ s.

44) [T] The formula for the volume of a sphere is $$S=\frac{4}{3}πr^3$$, where $$r$$ (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

a. Suppose $$r=\dfrac{1}{(t+1)^2}−\dfrac{1}{12}$$ where $$t$$ is time in minutes. Use the chain rule $$\dfrac{dS}{dt}=\dfrac{dS}{dr}⋅\dfrac{dr}{dt}$$ to find the rate at which the snowball is melting.

b. Use your result in part a. to find the rate at which the volume is changing at $$t=1$$ min.

a. $$\dfrac{dS}{dt}=−\dfrac{8πr^2}{(t+1)^3} = −\dfrac{8π\left( \dfrac{1}{(t+1)^2}−\dfrac{1}{12} \right)^2}{(t+1)^3}$$

b. The volume is decreasing at a rate of $$−\frac{π}{36}\; \text{ft}^3$$/min

45) [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function $$T(x)=94−10\cos\left[\frac{π}{12}(x−2)\right]$$, where $$x$$ is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

46) [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function $$D(t)=5\sin\left(\frac{π}{6}t−\frac{7π}{6}\right)+8$$, where $$t$$ is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

$$~2.3$$ ft/hr