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: $2^{4x+2}⋅ln2+8x$