Problem Set: The Chain Rule

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

1. $y=3u-6, \,\,\, u=2x^2$

2. $y=6u^3, \,\,\, u=7x-4$

Show Solution

$\frac{dy}{dx} = 18u^2 \cdot 7=18(7x-4)^2 \cdot 7$

3. $y= \sin u, \,\,\, u=5x-1$

4. $y= \cos u, \,\,\, u=\dfrac{−x}{8}$

Show Solution

$\frac{dy}{dx} = −\sin u \cdot \frac{-1}{8}=−\sin (\frac{−x}{8}) \cdot \frac{-1}{8}$

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

6. $y=\sqrt{4u+3}, \,\,\, u=x^2-6x$

Show Solution

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

For each of the following exercises (7-14),

decompose each function in the form $y=f(u)$ and $u=g(x)$ , and
find $\frac{dy}{dx}$ as a function of $x$ .

7. $y=(3x-2)^6$

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

Show Solution

a. $f(u) = u^3, \, u=3x^2+1$ ;
b. $\frac{dy}{dx} = 18x(3x^2+1)^2$

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

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

Show Solution

a. $f(u)=u^7, \, u=\frac{x}{7}+\frac{7}{x}$ ;
b. $\frac{dy}{dx} = 7(\frac{x}{7}+\frac{7}{x})^6 \cdot (\frac{1}{7}-\frac{7}{x^2})$

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

12. $y= \csc (\pi x+1)$

Show Solution

a. $f(u)= \csc u, \, u=\pi x+1$ ;
b. $\frac{dy}{dx} = −\pi \csc (\pi x+1) \cdot \cot (\pi x+1)$

13. $y= \cot^2 x$

14. $y=-6 \sin^{-3} x$

Show Solution

a. $f(u)=-6u^{-3}, \, u= \sin x$ ;
b. $\frac{dy}{dx} = 18 \sin^{-4} x \cdot \cos x$

For the following exercises (15-24), find $\frac{dy}{dx}$ for each function.

15. $y=(3x^2+3x-1)^4$

16. $y=(5-2x)^{-2}$

Show Solution

$\frac{dy}{dx} = \frac{4}{(5-2x)^3}$

17. $y= \cos^3 (\pi x)$

18. $y=(2x^3-x^2+6x+1)^3$

Show Solution

$\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)$

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

20. $y=(\tan x+ \sin x)^{-3}$

Show Solution

$\frac{dy}{dx} = -3(\tan x+ \sin x)^{-4} \cdot (\sec^2 x+ \cos x)$

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

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

Show Solution

$\frac{dy}{dx} = -7 \cos (\cos 7x) \cdot \sin 7x$

23. $y=\sqrt{6+ \sec \pi x^2}$

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

Show Solution

$\frac{dy}{dx} = -12 \cot^2 (4x+1) \cdot \csc^2 (4x+1)$

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

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

Show Solution

$10\frac{3}{4}$

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 $f^{\prime}(4)$ .

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

Show Solution

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

29. [T] Find the equation of the tangent line to $y=\left(3x+\dfrac{1}{x}\right)^2$ 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 $y=\left(x-\dfrac{6}{x}\right)^8$ is horizontal.

Show Solution

$x= \pm \sqrt{6}$

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

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

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

32. $h(x)=f(g(x)); \,\,\, a=0$

Show Solution

33. $h(x)=g(f(x)); \,\,\, a=0$

34. $h(x)=(x^4+g(x))^{-2}; \,\,\, a=1$

Show Solution

$-\frac{1}{8}$

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

36. $h(x)=f(x+f(x)); \,\,\, a=1$

Show Solution

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

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

Show Solution

39. $h(x)=f(g(\sin x)); \, a=0$

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

velocity and
acceleration.
Using a. and b. is the train speeding up or slowing down?

Show Solution

a. $-\frac{200}{343}$ m/s;
b. $\frac{600}{2401} \, \text{m/s}^2$ ;
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:

$s(t)=-3 \cos \left(\pi t+\dfrac{\pi}{4}\right)$ .

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

Find the marginal cost $C^{\prime}(x)$ .
Use Leibniz’s notation for the chain rule, $\frac{dC}{dt}=\frac{dC}{dx} \cdot \frac{dx}{dt}$ , to find the rate with respect to time $t$ that the cost is changing.
Use b. to determine how fast costs are increasing when $t=2$ weeks. Include units with the answer.

Show Solution

a. $C^{\prime}(x)=0.0003x^2-0.04x+3$
b. $\frac{dC}{dt}=100 \cdot (0.0003x^2-0.04x+3)$
c. Approximately $90,300 per week

43. [T] The formula for the area of a circle is $A=\pi r^2$ , 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.

Suppose $r=2-\dfrac{100}{(t+7)^2}$ where $t$ is time in seconds. Use the chain rule $\frac{dA}{dt}=\frac{dA}{dr} \cdot \frac{dr}{dt}$ to find the rate at which the area is expanding.
Use 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 $S=\frac{4}{3}\pi r^3$ , where $r$ (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

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

Show Solution

a. $\frac{dS}{dt}=-\frac{8\pi r^2}{(t+1)^3}$
b. The volume is decreasing at a rate of $-\frac{\pi}{36} \, \text{ft}^3/\text{min}$ .

45. [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function $T(x)=94-10 \cos \left[\dfrac{\pi}{12}(x-2)\right]$ , 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 $D(t)=5 \sin \left(\dfrac{\pi}{6} t-\dfrac{7\pi}{6}\right)+8$ , where $t$ is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

Show Solution

$\approx 2.3$ ft/hr

Derivatives: Supplemental Content

Problem Set: The Chain Rule

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