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

188 Chapter 3 Differentiation Rules
3.2 Exercises
1. Find the derivative of f sxd − s1 1 2x 2 dsx 2 x 2 d in two ways:
by using the Product Rule and by performing the multiplication
first. Do your answers agree?
2. Find the derivative of the function
Fsxd − x 4 2 5x 3 1 sx
x 2
in two ways: by using the Quotient Rule and by simplifying
first. Show that your answers are equivalent. Which method do
you prefer?
3–26 Differentiate.
3. f sxd − s3x 2 2 5xde x 4. tsxd − (x 1 2sx ) e x
5. y − x e x 6. y − e x
7. tsxd − 1 1 2x
3 2 4x
9. Hsud − su 2 su dsu 1 su d
10. Jsvd − sv 3 2 2vdsv 24 1 v 22 d
11. Fsyd −S 1 y 2 2 3 y 4Dsy 1 5y 3 d
12. f szd − s1 2 e z dsz 1 e z d
13. y − x 2 1 1
x 3 2 1
15. y − t 3 1 3t
t 2 2 4t 1 3
1 2 e x
8. Gsxd − x 2 2 2
2x 1 1
14. y − sx
2 1 x
16. y −
1
t 3 1 2t 2 2 1
17. y − e p sp 1 psp d 18. hsrd − ae r
b 1 e r
19. y − s 2 ss
20. y − sz 2 1 e z dsz
s 2
21. f std − s3 t
t 2 3
22. Vstd − 4 1 t
te t
23. f sxd − x 2 e x
x 2 1 e 24. Fstd − At
x Bt 2 1 Ct 3
25. f sxd − x
x 1 c x
27–30 Find f 9sxd and f99sxd.
26. f sxd − ax 1 b
cx 1 d
27. f sxd − sx 3 1 1de x 28. f sxd − sx e x
;
;
;
;
;
;
29. f sxd − x 2
x
30. f sxd −
x
1 1 e x 2 2 1
31–32 Find an equation of the tangent line to the given curve at
the specified point.
31. y − x 2 2 1
, s1, 0d
x 2 1 x 1 1 32.
y − 1 1 x
1 1 e x , (0, 1 2)
33–34 Find equations of the tangent line and normal line to the
given curve at the specified point.
33. y − 2xe x , s0, 0d 34. y − 2x , s1, 1d
x 2 1 1
35. (a) The curve y − 1ys1 1 x 2 d is called a witch of Maria
Agnesi. Find an equation of the tangent line to this curve
at the point s21, 1 2 d.
(b) Illustrate part (a) by graphing the curve and the tangent
line on the same screen.
36. (a) The curve y − xys1 1 x 2 d is called a serpentine. Find
an equation of the tangent line to this curve at the point
s3, 0.3d.
(b) Illustrate part (a) by graphing the curve and the tangent
line on the same screen.
37. (a) If f sxd − sx 3 2 xde x , find f 9sxd.
(b) Check to see that your answer to part (a) is reasonable by
comparing the graphs of f and f 9.
38. (a) If f sxd − e x ys2x 2 1 x 1 1d, find f 9sxd.
(b) Check to see that your answer to part (a) is reasonable by
comparing the graphs of f and f 9.
39. (a) If f sxd − sx 2 2 1dysx 2 1 1d, find f 9sxd and f 99sxd.
(b) Check to see that your answers to part (a) are reasonable
by comparing the graphs of f , f 9, and f99.
40. (a) If f sxd − sx 2 2 1de x , find f 9sxd and f 99sxd.
(b) Check to see that your answers to part (a) are reasonable
by comparing the graphs of f , f 9, and f 99.
41. If f sxd − x 2 ys1 1 xd, find f 99s1d.
42. If tsxd − xye x , find t snd sxd.
43. Suppose that f s5d − 1, f 9s5d − 6, ts5d − 23, and t9s5d − 2.
Find the following values.
(a) s ftd9s5d (b) s fytd9s5d (c) styf d9s5d
44. Suppose that f s4d − 2, ts4d − 5, f 9s4d − 6, and t9s4d − 23.
Find h9s4d.
(a) hsxd − 3f sxd 1 8tsxd (b) hsxd − f sxdtsxd
(c) hsxd − f sxd
tsxd
(d) hsxd −
tsxd
f sxd 1 tsxd
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