Problem Set: Derivatives of Trigonometric Functions

For the following exercises (1-10), find $\frac{dy}{dx}$ for the given functions.

1. $y=x^2- \sec x+1$

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$\frac{dy}{dx}=2x- \sec x \tan x$

2. $y=3 \csc x+\dfrac{5}{x}$

3. $y=x^2 \cot x$

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$\frac{dy}{dx}=2x \cot x-x^2 \csc^2 x$

4. $y=x-x^3 \sin x$

5. $y=\dfrac{\sec x}{x}$

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$\frac{dy}{dx}=\frac{x \sec x \tan x- \sec x}{x^2}$

6. $y= \sin x \tan x$

7. $y=(x+ \cos x)(1- \sin x)$

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$\frac{dy}{dx}=(1- \sin x)(1- \sin x)- \cos x(x+ \cos x)$

8. $y=\dfrac{\tan x}{1- \sec x}$

9. $y=\dfrac{1- \cot x}{1+ \cot x}$

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

10. $y= \cos x(1+ \csc x)$

For the following exercises (11-16), find the equation of the tangent line to each of the given functions at the indicated values of $x$ . Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

11. [T] $f(x)=−\sin x, \,\,\, x=0$

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$y=−x$

The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.

12. [T] $f(x)= \csc x, \,\,\, x=\frac{\pi}{2}$

13. [T] $f(x)=1+ \cos x, \,\,\, x=\frac{3\pi}{2}$

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

The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.

14. [T] $f(x)= \sec x, \,\,\, x=\frac{\pi}{4}$

15. [T] $f(x)=x^2- \tan x, \,\,\, x=0$

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$y=−x$

The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.

16. [T] $f(x)=5 \cot x, \,\,\, x=\frac{\pi}{4}$

For the following exercises (17-22), find $\frac{d^2 y}{dx^2}$ for the given functions.

17. $y=x \sin x- \cos x$

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$\frac{d^2 y}{dx^2} = 3 \cos x-x \sin x$

18. $y= \sin x \cos x$

19. $y=x-\frac{1}{2} \sin x$

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$\frac{d^2 y}{dx^2} = \frac{1}{2} \sin x$

20. $y=\frac{1}{x}+ \tan x$

21. $y=2 \csc x$

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$\frac{d^2 y}{dx^2} = 2\csc x( \csc^2 x + \cot^2 x)$

22. $y=\sec^2 x$

23. Find all $x$ values on the graph of $f(x)=-3 \sin x \cos x$ where the tangent line is horizontal.

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$x = \frac{(2n+1)\pi}{4}$ , where $n$ is an integer

24. Find all $x$ values on the graph of $f(x)=x-2 \cos x$ for [latex]0

25. Let $f(x)= \cot x$ . Determine the point(s) on the graph of $f$ for [latex]0

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$(\frac{\pi}{4},1), \, (\frac{3\pi}{4},-1)$

26. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $s(t)=-6 \cos t$ where $s$ is measured in inches and $t$ is measured in seconds. Find the rate at which the spring is oscillating at $t=5$ s.

27. Let the position of a swinging pendulum in simple harmonic motion be given by $s(t)=a \cos t+b \sin t$ where $a$ and $b$ are constants, $t$ measures time in seconds, and $s$ measures position in centimeters, If the position is $0$ cm and the velocity is $3$ cm/s when $t=0$ , find the values of $a$ and $b$ .

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$a=0, \, b=3$

28. After a diver jumps off a diving board, the edge of the board oscillates with position given by $s(t)=-5 \cos t$ cm at $t$ seconds after the jump.

Sketch one period of the position function for $t\ge 0$ .
Find the velocity function.
Sketch one period of the velocity function for $t\ge 0$ .
Determine the times when the velocity is 0 over one period.
Find the acceleration function.
Sketch one period of the acceleration function for $t\ge 0$ .

29. The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5 \sin x$ where $y$ is the number of hamburgers sold and $x$ represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $y^{\prime}$ and determine the intervals where the number of burgers being sold is increasing.

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$y^{\prime}=5 \cos (x)$ , increasing on $(0,\frac{\pi}{2}), \, (\frac{3\pi}{2},\frac{5\pi}{2})$ , and $(\frac{7\pi}{2},12)$

30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $y(t)=0.5+0.3 \cos t$ , where $t$ is the number of months since January. Find $y^{\prime}$ and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises (31-33), use the quotient rule to derive the given equations.

31. $\frac{d}{dx}(\cot x)=−\csc^2 x$

32. $\frac{d}{dx}(\sec x)= \sec x \tan x$

33. $\frac{d}{dx}(\csc x)=−\csc x \cot x$

34. Use the definition of derivative and the identity

$\cos (x+h)= \cos x \cos h- \sin x \sin h$ to prove that $\frac{d}{dx}(\cos x)=−\sin x$ .

For the following exercises (35-39), find the requested higher-order derivative for the given functions.

35. $\frac{d^3 y}{dx^3}$ of $y=3 \cos x$

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$\frac{d^3 y}{dx^3} = 3 \sin x$

36. $\frac{d^2 y}{dx^2}$ of $y=3 \sin x+x^2 \cos x$

37. $\frac{d^4 y}{dx^4}$ of $y=5 \cos x$

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$\frac{d^4 y}{dx^4} = 5 \cos x$

38. $\frac{d^2 y}{dx^2}$ of $y= \sec x+ \cot x$

39. $\frac{d^3 y}{dx^3}$ of $y=x^{10}- \sec x$

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$\frac{d^3 y}{dx^3} = 720x^7-5 \tan (x) \sec^3 (x)- \tan^3 (x) \sec (x)$

Derivatives: Supplemental Content

Problem Set: Derivatives of Trigonometric Functions

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