Derivatives of trigonometric functions example problems

For the following exercises (1-10), find [latex]\frac[/latex] for the given functions.

1. [latex]y=x^2- \sec x+1[/latex]

Show Solution [latex]\frac=2x- \sec x \tan x[/latex]

2. [latex]y=3 \csc x+\dfrac[/latex]

3. [latex]y=x^2 \cot x[/latex]

Show Solution

[latex]\frac=2x \cot x-x^2 \csc^2 x[/latex]

4. [latex]y=x-x^3 \sin x[/latex]

5. [latex]y=\dfrac[/latex]

Show Solution

6. [latex]y= \sin x \tan x[/latex]

7. [latex]y=(x+ \cos x)(1- \sin x)[/latex]

Show Solution

[latex]\frac=(1- \sin x)(1- \sin x)- \cos x(x+ \cos x)[/latex]

8. [latex]y=\dfrac[/latex]

Show Solution

10. [latex]y= \cos x(1+ \csc x)[/latex]

For the following exercises (11-16), find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[/latex]. 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] [latex]f(x)=−\sin x, \,\,\, x=0[/latex]

Show Solution

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

12. [T] [latex]f(x)= \csc x, \,\,\, x=\frac<\pi>[/latex]

13. [T] [latex]f(x)=1+ \cos x, \,\,\, x=\frac<3\pi>[/latex]

Show Solution

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] [latex]f(x)= \sec x, \,\,\, x=\frac<\pi>[/latex]

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

Show Solution

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] [latex]f(x)=5 \cot x, \,\,\, x=\frac<\pi>[/latex]

For the following exercises (17-22), find [latex]\frac[/latex] for the given functions.

17. [latex]y=x \sin x- \cos x[/latex]

Show Solution

[latex]\frac = 3 \cos x-x \sin x[/latex]

18. [latex]y= \sin x \cos x[/latex]

19. [latex]y=x-\frac \sin x[/latex]

Show Solution

20. [latex]y=\frac+ \tan x[/latex]

21. [latex]y=2 \csc x[/latex]

Show Solution

[latex]\frac = 2\csc x( \csc^2 x + \cot^2 x)[/latex]

22. [latex]y=\sec^2 x[/latex]

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

Show Solution

[latex]x = \frac<(2n+1)\pi>[/latex], where [latex]n[/latex] is an integer

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

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

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

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

Show Solution

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

  1. Sketch one period of the position function for [latex]t\ge 0[/latex].
  2. Find the velocity function.
  3. Sketch one period of the velocity function for [latex]t\ge 0[/latex].
  4. Determine the times when the velocity is 0 over one period.
  5. Find the acceleration function.
  6. Sketch one period of the acceleration function for [latex]t\ge 0[/latex].

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

Show Solution

[latex]y^<\prime>=5 \cos (x)[/latex], increasing on [latex](0,\frac<\pi>), \, (\frac<3\pi>,\frac<5\pi>)[/latex], and [latex](\frac<7\pi>,12)[/latex]

30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \cos t[/latex], where [latex]t[/latex] is the number of months since January. Find [latex]y^<\prime>[/latex] 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. [latex]\frac(\cot x)=−\csc^2 x[/latex]

32. [latex]\frac(\sec x)= \sec x \tan x[/latex]

33. [latex]\frac(\csc x)=−\csc x \cot x[/latex]

34. Use the definition of derivative and the identity

[latex]\cos (x+h)= \cos x \cos h- \sin x \sin h[/latex] to prove that [latex]\frac(\cos x)=−\sin x[/latex].

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

35. [latex]\frac[/latex] of [latex]y=3 \cos x[/latex]

Show Solution [latex]\frac = 3 \sin x[/latex]

36. [latex]\frac[/latex] of [latex]y=3 \sin x+x^2 \cos x[/latex]

37. [latex]\frac[/latex] of [latex]y=5 \cos x[/latex]

Show Solution

[latex]\frac = 5 \cos x[/latex]

38. [latex]\frac[/latex] of [latex]y= \sec x+ \cot x[/latex]

39. [latex]\frac[/latex] of [latex]y=x^- \sec x[/latex]

Show Solution

[latex]\frac = 720x^7-5 \tan (x) \sec^3 (x)- \tan^3 (x) \sec (x)[/latex]

Licenses and Attributions CC licensed content, Shared previously