1,001 Calculus Practice Problems

Part I

The Questions

 

Chapter 5

The Product, Quotient, and Chain Rules

This chapter focuses on some of the major techniques needed to find the derivative: the product rule, the quotient rule, and the chain rule. By using these rules along with the power rule and some basic formulas (see Chapter 4), you can find the derivatives of most of the single-variable functions you encounter in calculus. However, after using the derivative rules, you often need many algebra steps to simplify the function so that it's in a nice final form, especially on problems involving the product rule or quotient rule.

The Problems You'll Work On

Here you practice using most of the techniques needed to find derivatives (besides the power rule):

·        The product rule

·        The quotient rule

·        The chain rule

·        Derivatives involving trigonometric functions

What to Watch Out For

Many of these problems require one calculus step and then many steps of algebraic simplification to get to the final answer. Remember the following tips as you work through the problems:

·        Considering simplifying a function before taking the derivative. Simplifying before taking the derivative is almost always easier than finding the derivative and then simplifying.

·        Some problems have functions without specified formulas in the questions; don't be thrown off! Simply proceed as you normally would on a similar example.

·        Many people make the mistake of using the product rule when they should be using the chain rule. Stop and examine the function before jumping in and taking the derivative. Make sure you recognize whether the question involves a product or a composition (in which case you must use the chain rule).

·        Rewriting the function by adding parentheses or brackets may be helpful, especially on problems that involve using the chain rule multiple times.

Using the Product Rule to Find Derivatives

312–331 Use the product rule to find the derivative of the given function.

312. f (x) = (2x3 + 1)(x5 – x)

313. f (x) = xsin x

314. f (x) = sec x tan x

315. 9781118496718-eq05001.eps

316. f (x) = 4x csc x

317. Find (fg)'(4) if f (4) = 3, f '(4) = 2, g(4) = –6, and g '(4) = 8.

318. 9781118496718-eq05002.eps

319. f (x) = (sec x)(x + tan x)

320. 9781118496718-eq05003.eps

321. f (x) = 4x3 sec x

322. 9781118496718-eq05004.eps

323. Assuming that g is a differentiable function, find an expression for the derivative of f (x) = x2g(x).

324. Assuming that g is a differentiable function, find an expression for the derivative of 9781118496718-eq05005.eps.

325. Find (fg)'(3) if f (3) = –2, f '(3) = 4, g(3) = –8, and g '(3) = 7.

326. f (x) = x2 cos x sin x

327. Assuming that g is a differentiable function, find an expression for the derivative of 9781118496718-eq05006.eps.

328. 9781118496718-eq05007.eps

329. 9781118496718-eq05008.eps

330. Assuming that g is a differentiable function, find an expression for the derivative of 9781118496718-eq05009.eps.

331. Assuming that g and h are differentiable functions, find an expression for the derivative of 9781118496718-eq05010.eps.

Using the Quotient Rule to Find Derivatives

332–351 Use the quotient rule to find the derivative.

332. 9781118496718-eq05011.eps

333. 9781118496718-eq05012.eps

334. 9781118496718-eq05013.eps

335. 9781118496718-eq05014.eps

336. Assuming that f and g are differentiable functions, find the value of 9781118496718-eq05015.eps if f (4) = 5, f '(4) = –7, g(4) = 8, and g'(4) = 4.

337. 9781118496718-eq05016.eps

338. 9781118496718-eq05017.eps

339. 9781118496718-eq05018.eps

340. 9781118496718-eq05019.eps

341. 9781118496718-eq05020.eps

342. 9781118496718-eq05021.eps

343. Assuming that f and g are differentiable functions, find the value of 9781118496718-eq05022.eps if f (5) = –4, f '(5) = 2, g(5) = –7, and g'(5) = –6.

344. 9781118496718-eq05023.eps

345. 9781118496718-eq05024.eps

346. 9781118496718-eq05025.eps

347. 9781118496718-eq05026.eps

348. 9781118496718-eq05027.eps

349. 9781118496718-eq05028.eps

350. Assuming that g is a differentiable function, find an expression for the derivative of 9781118496718-eq05029.eps.

351. Assuming that g is a differentiable function, find an expression for the derivative of 9781118496718-eq05030.eps.

Using the Chain Rule to Find Derivatives

352–370 Use the chain rule to find the derivative.

352. 9781118496718-eq05031.eps

353. f (x) = sin(4x)

354. 9781118496718-eq05032.eps

355. 9781118496718-eq05033.eps

356. 9781118496718-eq05034.eps

357. 9781118496718-eq05035.eps

358. 9781118496718-eq05036.eps

359. f (x) = cos(x sin x)

360. 9781118496718-eq05037.eps

361. 9781118496718-eq05038.eps

362. f (x) = secx + tanx

363. 9781118496718-eq05039.eps

364. 9781118496718-eq05040.eps

365. 9781118496718-eq05041.eps

366. 9781118496718-eq05042.eps

367. 9781118496718-eq05043.eps, where x > 1

368. 9781118496718-eq05044.eps

369. 9781118496718-eq05045.eps

370. 9781118496718-eq05046.eps

More Challenging Chain Rule Problems

371–376 Solve the problem related to the chain rule.

371. Find all x values in the interval [0, 2π] where the function f (x) = 2 cos x + sinx has a horizontal tangent line.

372. Suppose that H is a function such that 9781118496718-eq05047.eps for x > 0. Find an expression for the derivative of 9781118496718-eq05048.eps.

373. Let 9781118496718-eq05049.eps, g(2) = –2, g'(2) = 4, f'(2) = 5, and f'(–2) = 7. Find the value of F'(2).

374. Let 9781118496718-eq05050.eps, f (2) = –2, f'(2) = –5, and f'(–2) = 8. Find the value of F'(2).

375. Suppose that H is a function such that 9781118496718-eq05051.eps for x > 0. Find an expression for the derivative of f (x) = H(x3).

376. Let 9781118496718-eq05052.epsg(4) = 6, g'(4) = 8, f '(4) = 2, and f '(6) = 10. Find the value of F '(4).