# 3.5E: Trig Derivatives Exercises

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## 3.5: Derivatives of Trigonometric Functions

### Exercise:

For the following exercises, find $$\frac{dy}{dx}$$ for the given functions.

175) $$y=x^2−secx+1$$

$$\frac{dy}{dx}=2x−secxtanx$$

176) $$y=3cscx+\frac{5}{x}$$

177) $$y=x^2cotx$$

$$\frac{dy}{dx}=2xcotx−x^2csc^2x$$

178) $$y=x−x^3sinx$$

179) $$y=\frac{secx}{x}$$

$$\frac{dy}{dx}=\frac{xsecxtanx−secx}{x^2}$$

180) $$y=sinxtanx$$

181) $$y=(x+cosx)(1−sinx)$$

$$\frac{dy}{dx}=(1−sinx)(1−sinx)−cosx(x+cosx)$$

182) $$y=\frac{tanx}{1−secx}$$

183) $$y=\frac{1−cotx}{1+cotx}$$

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

184) $$y=cosx(1+cscx)$$

For the following exercises, 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.

185) $$[T] f(x)=−\sin{x},x=0$$

$$y=−x$$

186) $$[T] f(x)=cscx,x=\frac{π}{2}$$

187) $$[T] f(x)=1+cosx,x=\frac{3π}{2}$$

$$y=x+\frac{2−3π}{2}$$

188) $$[T] f(x)=secx,x=\frac{π}{4}$$

189) $$[T] f(x)=x^2−\tan{x}=0$$

$$y=−x$$

190) $$[T] f(x)=5cotxx=\frac{π}{4}$$

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

191) $$y=xsinx−cosx$$

$$3cosx−xsinx$$

192) $$y=sinxcosx$$

193) $$y=x−\frac{1}{2}sinx$$

$$\frac{1}{2}sinx$$

194) $$y=\frac{1}{x}+tanx$$

195) $$y=2cscx$$

$$csc(x)(3csc^2(x)−1+cot^2(x))$$

196) $$y=sec^2x$$

197) Find all $$x$$ values on the graph of $$f(x)=−3sinxcosx$$ where the tangent line is horizontal.

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

198) Find all $$x$$ values on the graph of $$f(x)=x−2cosx$$ for $$0<x<2π$$ where the tangent line has slope 2.

199) Let $$f(x)=cotx.$$ Determine the points on the graph of $$f$$ for $$0<x<2π$$ where the tangent line(s) is (are) parallel to the line $$y=−2x$$.

$$(\frac{π}{4},1),(\frac{3π}{4},−1)$$

200) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $$s(t)=−6cost$$ 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.

201) Let the position of a swinging pendulum in simple harmonic motion be given by $$s(t)=acost+bsint$$. Find the constants $$a$$ and $$b$$ such that when the velocity is 3 cm/s, $$s=0$$ and $$t=0$$.

$$a=0,b=3$$

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

a. Sketch one period of the position function for $$t≥0$$.

b. Find the velocity function.

c. Sketch one period of the velocity function for $$t≥0$$.

d. Determine the times when the velocity is 0 over one period.

e. Find the acceleration function.

f. Sketch one period of the acceleration function for $$t≥0$$.

203) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $$y=10+5sinx$$ 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 $$y'$$ and determine the intervals where the number of burgers being sold is increasing.

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

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

For the following exercises, use the quotient rule to derive the given equations.

205) $$\frac{d}{dx}(cotx)=−csc^2x$$

206) $$\frac{d}{dx}(secx)=secxtanx$$

207) $$\frac{d}{dx}(cscx)=−cscxcotx$$

208) Use the definition of derivative and the identity $$cos(x+h)=cosxcosh−sinxsinh$$

to prove that $$\frac{d(cosx)}{dx}=−sinx$$.

For the following exercises, find the requested higher-order derivative for the given functions.

209) $$\frac{d^3y}{dx^3}$$ of $$y=3cosx$$

$$3sinx$$

210) $$\frac{d^2y}{dx^2}$$ of $$y=3sinx+x^2cosx$$

211) $$\frac{d^4y}{dx^4}$$ of $$y=5cosx$$

$$5cosx$$

212) $$\frac{d^2y}{dx^2}$$ of $$y=secx+cotx$$

213) $$\frac{d^3y}{dx^3}$$ of $$y=x^{10}−secx$$

$$720x^7−5tan(x)sec^3(x)−tan^3(x)sec(x)$$