James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

204 Chapter 3 Differentiation Rules
so
Dy
Dx − f f 9sbd 1 « 2gft9sad 1 « 1 g
As Dx l 0, Equation 8 shows that Du l 0. So both « 1 l 0 and « 2 l 0 as Dx l 0.
Therefore
dy
dx − lim Dy
Dx l 0 Dx − lim f f 9sbd 1 « 2gft9sad 1 « 1 g
Dx l 0
− f 9sbd t9sad − f 9stsadd t9sad
This proves the Chain Rule.
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3.4
Exercises
1–6 Write the composite function in the form f stsxdd.
[Identify the inner function u − tsxd and the outer function
y − f sud.] Then find the derivative dyydx.
1. y − s 3 1 1 4x 2. y − s2x 3 1 5d 4
3. y − tan x
5. y − e sx
7–46 Find the derivative of the function.
4. y − sinscot xd
6. y − s2 2 e x
7. Fsxd − s5x 6 1 2x 3 d 4 8. Fsxd − s1 1 x 1 x 2 d 99
9. f sxd − s5x 1 1
1
10. f sxd −
s 3 x 2 2 1
11. f sd − coss 2 d 12. tsd − cos 2
13. y − x 2 e 23x
14. f std − t sin t
15. f std − e at sin bt 16. tsxd − e x 2 2x
17. f sxd − s2x 2 3d 4 sx 2 1 x 1 1d 5
18. tsxd − sx 2 1 1d 3 sx 2 1 2d 6
19. hstd − st 1 1d 2y3 s2t 2 2 1d 3
20. Fstd − s3t 2 1d 4 s2t 1 1d 23
21. y −Î x
x 1 1
22. y −Sx 1 1 xD5
23. y − e tan 24. f std − 2 t 3
25. tsud −S u 3 8
2 1
u 3 1 1D
27. rstd − 10 2st
29. Hsrd − sr 2 2 1d 3
s2r 1 1d 5
26. sstd −Î 1 1 sin t
1 1 cos t
28. f szd − e zysz21d
30. Jsd − tan2 snd
;
t sin 2t
31. Fstd − e 32.
t 2
Fstd −
st 3 1 1
5
y 2 1 1D
33. Gsxd − 4 Cyx 34. Usyd −S y 4 1 1
35. y − cosS 1 2 e2x
1 1 e 2xD 36. y − x 2 e 21yx
37. y − cot 2 ssin d
38. y − s1 1 xe 22x
39. f std − tanssecscos tdd 40. y − e sin 2x 1 sinse 2x d
41. f std − sin 2 se sin 2t d 42. y − sx 1 sx 1 s x
43. tsxd − s2ra rx 1 nd p
44. y − 2 34x
45. y − cosssinstan xd 46. y − fx 1 sx 1 sin 2 xd 3 g 4
47–50 Find y9 and y99.
47. y − cosssin 3d
49. y − s1 2 sec t
48. y −
50. y − e ex
1
s1 1 tan xd 2
51–54 Find an equation of the tangent line to the curve at the given
point.
51. y − 2 x , s0, 1d 52. y − s1 1 x 3 , s2, 3d
53. y − sinssin xd, s, 0d 54. y − xe 2x 2 , s0, 0d
55. (a) Find an equation of the tangent line to the curve
y − 2ys1 1 e 2x d at the point s0, 1d.
(b) Illustrate part (a) by graphing the curve and the tangent line
on the same screen.
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