Chapter 3 | Derivatives
297
3.6 EXERCISES For the following exercises, given y = f ( u ) and u = g ( x ), find by using Leibniz’s notation for the
230. y =cos 3 ( πx )
dy dx dy du
3
⎛ ⎝ 2 x 3 − x 2 +6 x +1 ⎞ ⎠
231. y =
dy dx =
du dx .
chain rule:
232. y = 1 sin 2 ( x ) 233. y = (tan x +sin x ) −3 234. y = x 2 cos 4 x 235. y = sin(cos7 x ) 236. y = 6+sec πx 2 237. y =cot 3 (4 x +1) 238. Let y = ⎡ ⎣ f ( x ) ⎤
214. y =3 u −6, u =2 x 2 215. y =6 u 3 , u =7 x −4 216. y = sin u , u =5 x −1 217. y =cos u , u = − x 8 218. y = tan u , u =9 x +2 219. y = 4 u +3, u = x 2 −6 x
⎦ 3 and suppose that f ′(1) =4 and
For each of the following exercises, a. decompose each function in the form y = f ( u ) and u = g ( x ), and
dy dx =10
for x =1. Find f (1).
4
dy dx
⎛ ⎝ f ( x )+5 x 2 ⎞ ⎠
as a function of x .
y =
b. find
239.
Let
and suppose that
dy dx =3
f (−1) =−4 and
when x =−1. Find f ′(−1)
220. y = (3 x −2) 6
⎠ 2 and u = x 3 −2 x . If
Let y = ⎛
⎝ f ( u )+3 x ⎞
240.
3
⎛ ⎝ 3 x 2 +1 ⎞ ⎠
221. y =
dy dx =18
f (4) =6 and
when x =2, find f ′(4).
222. y = sin 5 ( x )
241. [T] Find the equation of the tangent line to y =−sin ⎛ ⎝ x 2 ⎞ ⎠ at the origin. Use a calculator to graph the function and the tangent line together. 242. [T] Find the equation of the tangent line to y = ⎛ ⎝ 3 x + 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 − 6 x ⎞ ⎠ 8 is horizontal. 244. [T] Find an equation of the line that is normal to g ( θ ) = sin 2 ( πθ ) at the point ⎛ ⎝ 1 4 , 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
⎞ ⎠
7
7 x
223. y =
7 +
224. y = tan(sec x ) 225. y =csc( πx +1) 226. y =cot 2 x 227. y =−6sin −3 x
dy dx
For the following exercises, find
for each function.
4
⎛ ⎝ 3 x 2 +3 x −1 ⎞ ⎠
228. y =
229. y = (5−2 x ) −2
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