11DIFFERENTIATION - Department of Mathematics

730 11 DIFFERENTIATION
S ELF-CHECK E XERCISES 11.3
11.3 Exercises
1. Find the derivative of
In Exercises 1–46, find the derivative of the
given function.
1. f(x) (2x 1) 4 2. f(x) (1 x) 3
3. f(x) (x 2 2) 5 4. f(t) 2(t 3 1) 5
5. f(x) (2x x 2 ) 3 6. f(x) 3(x 3 x) 4
7. f(x) (2x 1) 2 8. f(t) 1
2 (2t2 t) 3
9. f(x) (x 2 4) 3/2 10. f(t) (3t 2 2t 1) 3/2
11. f(x) 3x 2 12. f(t) 3t 2 t
13. f(x) 3 1 x 2 14. f(x) 2x 2 2x 3
15. f(x)
1
(2x 3) 3
1
17. f(t)
2t 3
19. y
1
(4x 4 x) 3/2
21. f(x) (3x 2 2x 1) 2
22. f(t) (5t 3 2t 2 t 4) 3
23. f(x) (x 2 1) 3 (x 3 1) 2
24. f(t) (2t 1) 4 (2t 1) 4
16. f(x)
1
f(x)
2x2 1
2. Suppose the life expectancy at birth (in years) of a female in a certain country is
described by the function
g(t) 50.02(1 1.09t) 0.1 (0 t 150)
where t is measured in years and t 0 corresponds to the beginning of 1900.
a. What is the life expectancy at birth of a female born at the beginning of 1980?
At the beginning of the year 2000?
b. How fast is the life expectancy at birth of a female born at any time t changing?
Solutions to Self-CheckExercises 11.3 can be found on page 733.
2
(x 2 1) 4
1
18. f(x)
2x2 1
20. f(t)
4
3 2t 2 t
25. f(t) (t 1 t 2 ) 3 26. f(v) (v 3 4v 2 ) 3
27. f(x) x 1 x 1
28. f(u) (2u 1) 3/2 (u 2 1) 3/2
29. f(x) 2x 2 (3 4x) 4 30. h(t) t 2 (3t 4) 3
31. f(x) (x 1) 2 (2x 1) 4
32. g(u) (1 u 2 ) 5 (1 2u 2 ) 8
3
x 3
33. f(x) x 2
35. s(t) t
3/2
2t 1
u 1
37. g(u) 3u 2
39. f(x)
41. h(x) (3x2 1) 3
(x 2 1) 4
43. f(x)
5
x 1
34. f(x) x 1
36. g(s) s2 1
3/2
s
2x 1
38. g(x) 2x 1
x 2
(x 2 1) 4 40. g(u) 2u2
(u 2 u) 3
2x 1
x 2 1
t 1
45. g(t)
t 2 1
42. g(t)
(2t 1)2
(3t 2) 4
4t 2
44. f(t)
2t2 2t 1
46. f(x) x2 1
x 2 1