## ExercisesforSection 3: Derivatives of Trigonometric Functions

Find the derivatives of the following functions

 1. $f(x) = \sin x - \cos x$ 2. $f(x) = \tan x -\sin x$ 3. $g(x) = (\sin x)(\tan x)$ 4. $g(x) = (\cos x)(cotan x)$ 5. $h(x) = 2cosec x - \sec x + 3x$ 6. $h(x) = 2\sec x + 3\tan x + 3x$ 7. $r(x) = x \cos x + x^2 + 1$ 8. $r(x) = 2x \sin x - x^2$ 9. $s(x) = (x^2-x+1)\tan x$ 10. $s(x) = \frac{\tan x}{x^2-1}$ 11. $t(x) = \frac{cotan x}{1 + \sec x}$ 12. $t(x) = (1+\sec x)(1-\cos x)$ 13. $k(x) = \cos^2x$ 14. $k(x) = \tan^2x$ 15. $j(x) = \sec^2x$ 16. $j(x) = cosec^2x$ 17. $u(x) = \cos(x^2-x)$ 18. $u(x) = \sin(3x^2+x-1)$ 19. $v(x) = \sec(x^{2.2}+1.2x-1)$ 20. $v(x) = \tan(x^{2.2}+1.2x-1)$ 21. $w(x) = (\sec x)(\tan(x^2-1))$ 22. $w(x) = (\cos x)(\sec(x^2-1))$ 23. $y(x) = \cos(e^x) + e^x \cos x$ 24. $y(x) = \sec(e^x)$ 25. $z(x) = \ln \|\sec x + \tan x\|$ 26. $z(x) = \ln \|cosec x + cotan x\|$

Obtain the following derivatives

 27 $\frac{d}{dx} \tan x = \sec^2x$ 28 $\frac{d}{dx} cotan x = - cosec^2x$ 29 $\frac{d}{dx} \sec x = \sec^x \tan x$ 30 $\frac{d}{dx} \ln \|\cos x\| = - \tan x$

Calculate the following

 31 $\frac{d}{dx}[e^{-2x}\sin(3x)]$ 32 $\frac{d}{dx}[e^{5x} \sin(-4x)]$ 33 $\frac{d}{dx}[\sin(3x)]^{0.5}$
 34 $\frac{d}{dx} \cos \left(\frac{x^2}{x - 1}\right)$ 35 $\frac{d}{dx} \sec \left(\frac{x^3}{x^2 - 1} \right)$ 36 $\frac{d}{dx} \left(\frac{\tan x}{2 + e^x} \right)^2$
 37 $\frac{d}{dx}([\ln \|x\|][cotan(2x-1)])$ 38 $\frac{d}{dx} \ln \|\sin x-2xe^{-x}\|$

Applications

39. Cost
The cost of Dig-It brand snow shovels is given by

$c(t) = 3.5\sin[2π(t-0.75)],$

where t is time in years since January 1, 1997. How fast, in dollars per week, is the cost increasing each September 1?

40. Sales
Daily sales of Doggy brand cookies can be modeled by

$s(t) = 400\cos[2π(t-2/7)]$

cartons per week, where t is time in weeks since Monday morning. How fast are sales going up on Thursday morning?

41. Tides
The depth of water at my favorite surfing spot varies from $5$ft to $15$ft, depending on the time. Last Sunday, high tide occurred at 5:00 am. and the next high tide occurred at 6:30 pm.

(a) Obtain a cosine model describing the depth of water as a function of time t in hours since 5:00 am. on Sunday morning.

(b) How fast was the tide rising (or falling) at noon on Sunday?

42. Tides
Repeat Exercise 41 using data from the depth of water at my other favorite surfing spot, where the tide last Sunday varied from a low of $6$ft at 4:00 am to a high of $10$ft. at noon.

43. Inflated Cost
Taking a $3.5%$ rate of inflation into account, the cost of Dig-In brand snow shovels is given by

$c(t) = 1.035^t[0.8\sin(2πt) + 10.2],$

where $t$ is time in years since January 1, 1997.

How fast, in dollars per week, is the cost of Dig In shovels increasing on January 1, 1998?

44. Deflation
Sales (in bottles per day) of my exclusive 1997 vintage Chateau Petit Mont Blanc follow the function

$s(t) = 4.5e^{-0.2t}\sin(2πt),$

where t is time in years since January 1, 1997. How fast were sales rising or falling one year later?

Communication and Reasoning Exercises

45. Give two examples of a function $f(x)$ with the property that $f''(x) = -f(x).$

46. Give three examples of a function $f(x)$ with the property that $f^{(4)}(x) = f(x).$

47. By referring to the graph of $f(x) = \cos x,$ explain why $f'(x) = - \sin x,$ rather than $\sin x.$

48. At what angle does the graph of $f(x) = \sin x$ depart from the origin?

We would welcome comments and suggestions for improving this resource. Mail us at:
 Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: September, 1996