3.7 E: Chain Rule Exercises

3.7: The Chain Rule

Exercise:

For the following exercises, given \(y=f(u)\) and \(u=g(x)\), find dydx by using Leibniz’s notation for the chain rule: \(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.\)

214) \(y=3u−6,u=2x^2\)

215) \(y=6u^3,u=7x−4\)

Solution: \(18u^2⋅7=18(7x−4)^2⋅7\)

216) \(y=sinu,u=5x−1\)

217) \(y=cosu,u=\frac{−x}{8}\)

Solution: \(−sinu⋅\frac{−1}{8=}−sin(\frac{−x}{8})⋅\frac{−1}{8}\)

218) \(y=tanu,u=9x+2\)

219) \(y=\sqrt{4u+3},u=x^2−6x\)

Solution: \(\frac{8x−24}{2\sqrt{4u+3}}=\frac{4x−12}{\sqrt{4x^2−24x+3}}\)

For each of the following exercises,

a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and

b. find \(\frac{dy}{dx}\) as a function of \(x\).

220) \(y=(3x−2)^6\)

221) \(y=(3x^2+1)^3\)

Solution: a. \(u=3x^2+1\); b. \(18x(3x^2+1)^2\)

222) \(y=sin^5(x)\)

For each of the following exercises, find \(\frac{dy}{dx}\) as a function of \(x\).

223) \(y=(\frac{x}{7}+\frac{7}{x})^7\)

Solution: \(a. f(u)=u^7,u=\frac{x}{7}+\frac{7}{x}; b. 7(\frac{x}{7}+\frac{7}{x})^6⋅(\frac{1}{7}−\frac{7}{x^2})\)

224) \(y=tan(secx)\)

225) \(y=csc(πx+1)\)

Solution: \(a. f(u)=cscu,u=πx+1; b. −πcsc(πx+1)⋅cot(πx+1)\)

226) \(y=cot^2x\)

227) \(y=−6sin^{−3}x\)

a. \(f(u)=−6u^{−3},u=sinx, b. 18sin^{−4}x⋅cosx\)

For the following exercises, find \(\frac{dy}{dx}\) for each function.

228) \(y=(3x^2+3x−1)^4\)

229) \(y=(5−2x)^{−2}\)

Solution: \(\frac{4}{(5−2x)^3}\)

230) \(y=cos^3(πx)\)

231) \(y=(2x^3−x^2+6x+1)^3\)

Solution: \(6(2x^3−x^2+6x+1)^2(3x^2−x+3)\)

232) \(y=\frac{1}{sin^2(x)}\)

233) \(y=(tanx+sinx)^{−3}\)

Soution: \(−3(tanx+sinx)^{−4}⋅(sec^2x+cosx)\)

234) \(y=x^2cos^4x\)

235) \(y=sin(cos7x)\)

Solution: \(−7cos(cos7x)⋅sin7x\)

236) \(y=\sqrt{6+secπx^2}\)

237) \(y=cot^3(4x+1)\)

Solution: \(−12cot^2(4x+1)⋅csc^2(4x+1)\)

238) Let \(y=[f(x)]^3\) and suppose that \(f′(1)=4\) and \(\frac{dy}{dx}=10\) for \(x=1\). Find \(f(1)\).

239) Let \(y=(f(x)+5x^2)^4\) and suppose that \(f(−1)=−4\) and \(\frac{dy}{dx}=3\) when \(x=−1\). Find \(f′(−1)\)

Solution: \(10\frac{3}{4}\)

240) Let \(y=(f(u)+3x)^2\) and \(u=x^3−2x\). If \(f(4)=6\) and \(\frac{dy}{dx}=18\) when \(x=2\), find \(f′(4)\).

241) [T] Find the equation of the tangent line to \(y=−sin(\frac{x}{2})\) at the origin. Use a calculator to graph the function and the tangent line together.

Solution: \(y=\frac{−1}{2}x\)

242) [T] Find the equation of the tangent line to \(y=(3x+\frac{1}{x})^2\) at the point \((1,16)\). Use a calculator to graph the function and the tangent line together.

243) Find the \(x\) -coordinates at which the tangent line to \(y=(x−\frac{6}{x})^8\) is horizontal.

Solution: \(x=±\sqrt{6}\)

244) [T] Find an equation of the line that is normal to \(g(θ)=sin2^(πθ)\) at the point \((\frac{1}{4},\frac{1}{2})\). Use a calculator to graph the function and the normal line together.

For the following exercises, use the information in the following table to find \(h′(a)\) at the given value for \(a\).

\(x\)	\(f(x)\)	\(f'(x)\)	\(g(x)\)	\(g'(x)\)
0	2	5	0	2
1	1	−2	3	0
2	4	4	1	−1
3	3	−3	2	3

245) \(h(x)=f(g(x));a=0\)

Solution: \(10\)

246) \(h(x)=g(f(x));a=0\)

247) \(h(x)=(x^4+g(x))^{−2};a=1\)

Solution: \(−\frac{1}{8}\)

248) \(h(x)=(\frac{f(x)}{g(x)})^2;a=3\)

249) \(h(x)=f(x+f(x));a=1\)

Solution: \(−4\)

250) \(h(x)=(1+g(x))^3;a=2\)

251) \(h(x)=g(2+f(x^2));a=1\)

Solution: \(−12\)

252) h(x)=f(g(sinx));a=0

253) [T] The position function of a freight train is given by

\(s(t)=100(t+1)^{−2}\), with \(s\) in meters and \(t\) in seconds. At time \(t=6\)s, find the train’s

a. velocity and

b. acceleration.

c. Using a. and b. is the train speeding up or slowing down?

Solution: \(a. −\frac{200}{343}\) m/s, b. \(\frac{600}{2401}\) m/s^2, c. The train is slowing down since velocity and acceleration have opposite signs.

254) [T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and \(s\) is in inches:

\(s(t)=−3cos(πt+\frac{π}{4}).\)

a. Determine the position of the spring at \(t=1.5\) s.

b. Find the velocity of the spring at \(t=1.5\) s.

255) [T] The total cost to produce \(x\) boxes of Thin Mint Girl Scout cookies is \(C\) dollars, where \(C=0.0001x^3−0.02x^2+3x+300.\) In \(t\) weeks production is estimated to be \(x=1600+100t\) boxes.

a. Find the marginal cost \(C′(x).\)

b. Use Leibniz’s notation for the chain rule, \(\frac{dC}{dt}=\frac{dC}{dx}⋅\frac{dx}{dt}\), to find the rate with respect to time \(t\) that the cost is changing.

c. Use b. to determine how fast costs are increasing when \(t=2\) weeks. Include units with the answer.

Solution: \(a. C′(x)=0.0003x^2−0.04x+3\)

\(b. dCdt=100⋅(0.0003x^2−0.04x+3)\) c. Approximately $90,300 per week

256) [T] The formula for the area of a circle is \(A=πr^2\), where \(r\) is the radius of the circle. Suppose a circle is expanding, meaning that both the area \(A\) and the radius \(r\) (in inches) are expanding.

a. Suppose \(r=2−\frac{100}{(t+7)^2}\) where \(t\) is time in seconds. Use the chain rule \(\frac{dA}{dt}=\frac{dA}{dr}⋅\frac{dr}{dt}\) to find the rate at which the area is expanding.

b. Use a. to find the rate at which the area is expanding at \(t=4\) s.

257) [T] The formula for the volume of a sphere is \(S=\frac{4}{3}πr^3\), where \(r\) (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

a. Suppose \(r=\frac{1}{(t+1)^2}−\frac{1}{12}\) where t is time in minutes. Use the chain rule \(\frac{dS}{dt}=\frac{dS}{dr}⋅\frac{dr}{dt}\) to find the rate at which the snowball is melting.

b. Use a. to find the rate at which the volume is changing at \(t=1\) min.

Solution: \(a. \frac{dS}{dt}=−\frac{8πr^2}{(t+1)^3}\) b. The volume is decreasing at a rate of \(−\frac{π}{36}\) \(ft^3\)/min

258) [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function \(T(x)=94−10cos[\frac{π}{12}(x−2)]\), where \(x\) is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

259) [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function \(D(t)=5sin(\frac{π}{6}t−\frac{7π}{6})+8\), where \(t\) is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

Solution: \(~2.3\) ft/hr

More Exercises

For the following exercises, find \(f′(x)\) for each function.

331) \(f(x)=x^2e^x\)

Solution: \(2xe^x+x^2e^x\)

332) \(f(x)=\frac{e^{−x}}{x}\)

333) \(f(x)=e^{x^3lnx}\)

Solution: \(e^{x^3}lnx(3x^2lnx+x^2)\)

334) \(f(x)=\sqrt{e^{2x}+2x}\)

335) \(f(x)=\frac{e^x−e^{−x}}{e^x+e^{−x}}\)

Solution: \(\frac{4}{(e^x+e^{−x})^2}\)

336) \(f(x)=\frac{10^x}{ln10}\)

337) \(f(x)=2^{4x}+4x^2\)

Solution: \(2^{4x+2}⋅ln2+8x\)