Problem Set: Differentiation Rules

For the following exercises (1-12), find $f^{\prime}(x)$ for each function.

1. $f(x)=x^7+10$

2. $f(x)=5x^3-x+1$

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$f^{\prime}(x)=15x^2-1$

3. $f(x)=4x^2-7x$

4. $f(x)=8x^4+9x^2-1$

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$f^{\prime}(x)=32x^3+18x$

5. $f(x)=x^4+\dfrac{2}{x}$

6. $f(x)=3x\left(18x^4+\dfrac{13}{x+1}\right)$

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$f^{\prime}(x)=270x^4+\frac{39}{(x+1)^2}$

7. $f(x)=(x+2)(2x^2-3)$

8. $f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)$

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$f^{\prime}(x)=\frac{-5}{x^2}$

9. $f(x)=\dfrac{x^3+2x^2-4}{3}$

10. $f(x)=\dfrac{4x^3-2x+1}{x^2}$

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$f^{\prime}(x)=\frac{4x^4+2x^2-2x}{x^4}$

11. $f(x)=\dfrac{x^2+4}{x^2-4}$

12. $f(x)=\dfrac{x+9}{x^2-7x+1}$

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$f^{\prime}(x)=\frac{−x^2-18x+64}{(x^2-7x+1)^2}$

For the following exercises (13-16), find the equation of the tangent line $T(x)$ to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

13. [T] $y=3x^2+4x+1$ at $(0,1)$

14. [T] $y=2\sqrt{x}+1$ at $(4,5)$

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The graph y is a slightly curving line with y intercept at 1. The line T(x) is straight with y intercept 3 and slope 1/2.

$T(x)=\frac{1}{2}x+3$

15. [T] $y=\dfrac{2x}{x-1}$ at $(-1,1)$

16. [T] $y=\dfrac{2}{x}-\dfrac{3}{x^2}$ at $(1,-1)$

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The graph y is a two crescents with the crescent in the third quadrant sloping gently from (−3, −1) to (−1, −5) and the other crescent sloping more sharply from (0.8, −5) to (3, 0.2). The straight line T(x) is drawn through (0, −5) with slope 4.

$T(x)=4x-5$

For the following exercises (17-20), assume that $f(x)$ and $g(x)$ are both differentiable functions for all $x$ . Find the derivative of each of the functions $h(x)$ .

17. $h(x)=4f(x)+\dfrac{g(x)}{7}$

18. $h(x)=x^3f(x)$

Show Solution

$h^{\prime}(x)=3x^2f(x)+x^3f^{\prime}(x)$

19. $h(x)=\dfrac{f(x)g(x)}{2}$

20. $h(x)=\dfrac{3f(x)}{g(x)+2}$

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$h^{\prime}(x)=\frac{3f^{\prime}(x)(g(x)+2)-3f(x)g^{\prime}(x)}{(g(x)+2)^2}$

For the following exercises (21-24), assume that $f(x)$ and $g(x)$ are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

$x$	1	2	3	4
$f(x)$	3	5	-2	0
$g(x)$	2	3	-4	6
$f^{\prime}(x)$	-1	7	8	-3
$g^{\prime}(x)$	4	1	2	9

21. Find $h^{\prime}(1)$ if $h(x)=xf(x)+4g(x)$ .

22. Find $h^{\prime}(2)$ if $h(x)=\dfrac{f(x)}{g(x)}$ .

Show Solution

$\frac{16}{9}$

23. Find $h^{\prime}(3)$ if $h(x)=2x+f(x)g(x)$ .

24. Find $h^{\prime}(4)$ if $h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}$ .

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For the following exercises (25-27), use the following figure to find the indicated derivatives, if they exist.

Two functions are graphed: f(x) and g(x). The function f(x) starts at (−1, 5) and decreases linearly to (3, 1) at which point it increases linearly to (5, 3). The function g(x) starts at the origin, increases linearly to (2.5, 2.5), and then remains constant at y = 2.5.

25. Let $h(x)=f(x)+g(x)$ . Find

$h^{\prime}(1)$
$h^{\prime}(3)$
$h^{\prime}(4)$

26. Let $h(x)=f(x)g(x)$ . Find

$h^{\prime}(1)$
$h^{\prime}(3)$
$h^{\prime}(4)$

Show Solution

27. Let $h(x)=\dfrac{f(x)}{g(x)}$ . Find

$h^{\prime}(1)$
$h^{\prime}(3)$
$h^{\prime}(4)$

For the following exercises (28-31),

Evaluate $f^{\prime}(a)$ , and
Graph the function $f(x)$ and the tangent line at $x=a$ .

28. [T] $f(x)=2x^3+3x-x^2, \,\,\, a=2$

Show Solution

a. 23
b. $y=23x-28$

The graph is a slightly deformed cubic function passing through the origin. The tangent line is drawn through (0, −28) with slope 23.

29. [T] $f(x)=\dfrac{1}{x}-x^2, \,\,\, a=1$

30. [T] $f(x)=x^2-x^{12}+3x+2, \,\,\, a=0$

Show Solution

a. 3
b. $y=3x+2$

The graph starts in the third quadrant, increases quickly and passes through the x axis near −0.9, then increases at a lower rate, passes through (0, 2), increases to (1, 5), and then decreases quickly and passes through the x axis near 1.2.

31. [T] $f(x)=\dfrac{1}{x}-x^{\frac{2}{3}}, \,\,\, a=-1$

32. Find the equation of the tangent line to the graph of $f(x)=2x^3+4x^2-5x-3$ at $x=-1$ .

Show Solution

$y=-7x-3$

33. Find the equation of the tangent line to the graph of $f(x)=x^2+\dfrac{4}{x}-10$ at $x=8$ .

34. Find the equation of the tangent line to the graph of $f(x)=(3x-x^2)(3-x-x^2)$ at $x=1$ .

Show Solution

$y=-5x+7$

35. Find the point on the graph of $f(x)=x^3$ such that the tangent line at that point has an $x$ intercept of 6.

36. Find the equation of the line passing through the point $P(3,3)$ and tangent to the graph of $f(x)=\dfrac{6}{x-1}$ .

Show Solution

$y=-\frac{3}{2}x+\frac{15}{2}$

37. Determine all points on the graph of $f(x)=x^3+x^2-x-1$ for which

the tangent line is horizontal
the tangent line has a slope of $-1$

38. Find a quadratic polynomial such that $f(1)=5, \, f^{\prime}(1)=3$ , and $f''(1)=-6$ .

Show Solution

$y=-3x^2+9x-1$

39. A car driving along a freeway with traffic has traveled $s(t)=t^3-6t^2+9t$ meters in $t$ seconds.

Determine the time in seconds when the velocity of the car is 0.
Determine the acceleration of the car when the velocity is 0.

40. [T] A herring swimming along a straight line has traveled $s(t)=\dfrac{t^2}{t^2+2}$ feet in $t$ seconds.

Determine the velocity of the herring when it has traveled 3 seconds.

Show Solution

$\frac{12}{121}$ or 0.0992 ft/s

41. The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function $P(t)=\dfrac{8t+3}{0.2t^2+1}$ , where $t$ is measured in years.

Determine the initial flounder population.
Determine $P^{\prime}(10)$ and briefly interpret the result.

42. [T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C(t)=\dfrac{2t^2+t}{t^3+50}$ , where $C$ is measured in milligrams per liter of blood.

Find the rate of change of $C(t)$ .
Determine the rate of change for $t=8, \, 12, \, 24$ , and $36$ .
Briefly describe what seems to be occurring as the number of hours increases.

Show Solution

a. $\frac{-2t^4-2t^3+200t+50}{(t^3+50)^2}$
b. -0.02395 mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr
c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.

43. A book publisher has a cost function given by $C(x)=\dfrac{x^3+2x+3}{x^2}$ , where $x$ is the number of copies of a book in thousands and $C$ is the cost, per book, measured in dollars. Evaluate $C^{\prime}(2)$ and explain its meaning.

44. [T] According to Newton’s law of universal gravitation, the force $F$ between two bodies of constant mass $m_1$ and $m_2$ is given by the formula $F=\dfrac{G m_1 m_2}{d^2}$ , where $G$ is the gravitational constant and $d$ is the distance between the bodies.

Suppose that $G, \, m_1$ , and $m_2$ are constants. Find the rate of change of force $F$ with respect to distance $d$ .
Find the rate of change of force $F$ with gravitational constant $G=6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$ , on two bodies 10 meters apart, each with a mass of 1000 kilograms.

Show Solution

a. $F^{\prime}(d)=\frac{-2Gm_1 m_2}{d^3}$
b. $-1.33 \times 10^{-7}$ N/m

Derivatives: Supplemental Content

Problem Set: Differentiation Rules

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