Chapter 3 | Derivatives
331
3.9 EXERCISES For the following exercises, find f ′( x ) for each function. 331. f ( x ) = x 2 e x 332. f ( x ) = e − x x
ln x
⎛ ⎝ x 2 −1
⎞ ⎠
350. y =
351. y = x cot x 352. y = x +11 x 2 −4 3 353. y = x −1/2 ⎛
3 ln x
333. f ( x ) = e x
2/3
⎞ ⎠
⎝ x 2 +3
(3 x −4) 4
334. f ( x ) = e 2 x +2 x 335. f ( x ) = e x − e − x e x + e − x 336. f ( x ) = 10 x ln10 337. f ( x ) =2 4 x +4 x 2 338. f ( x ) =3 sin3x 339. f ( x ) = x π · π x 340. f ( x ) = ln ⎛ 341. f ( x ) = ln 5 x −7 342. f ( x ) = x 2 ln9 x 343. f ( x ) = log(sec x )
354. [T] Find an equation of the tangent line to the graph of f ( x ) =4 xe ⎛ ⎝ x 2 −1 ⎞ ⎠ at the point where x =−1. Graph both the function and the tangent line. 355. [T] Find the equation of the line that is normal to the graph of f ( x ) = x ·5 x at the point where x =1. Graph both the function and the normal line. 356. [T] Find the equation of the tangent line to the graph of x 3 − x ln y + y 3 =2 x +5 at the point where x =2. ( Hint : Use implicit differentiation to find dy dx .) Graph both the curve and the tangent line. 357. Consider the function y = x 1/ x for x >0. a. Determine the points on the graph where the tangent line is horizontal. b. Determine the points on the graph where y ′ >0 and those where y ′ <0.
⎝ 4 x 3 + x ⎞ ⎠
5
344. f ( x ) = log 7 ⎛
⎝ 6 x 4 +3 ⎞ ⎠
x 2 −4
345. f ( x ) =2 x · log
3 7
For the following exercises, use logarithmic differentiation to find dy dx . 346. y = x x
347. y = (sin2 x ) 4 x 348. y = (ln x ) ln x
log 2 x
349. y = x
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