5128_Ch03_pp098-184

Section 3.6 Chain Rule 153
Quick Review 3.6
(For help, go to Sections 1.2 and 1.6.)
In Exercises 1–5, let f x sin x, gx x 2 1, and hx 7x.
Write a simplified expression for the composite function.
1. f gx sin (x 2 1) 2. f ghx sin (49x 2 1)
3. g hx 49x 2 1 4. h gx 7x 2 7
5. f ( g x
hx
)
sin x2 7
1
x
In Exercises 6–10, let f x cos x, gx x 2, and
hx 3x 2 . Write the given function as a composite of two or more
of f, g, and h. For example, cos 3x 2 is f hx.
6. cosx 2 g(f(x)) 7. 3 cos 2 x 2 g(h(f(x)))
8. 3 cos x 6 h(g(f(x))) 9. cos 27x 4 f(h(h(x)))
10. cos 2 3x 2 f(g(h(x)))
3. 3 sin (3x)
4. (2 3x 2 ) sec 2 (2x – x 3 )
Section 3.6 Exercises
6. 1 0
x
2 csc 2 2 x 9. 3 sin 2 3t 13. 2(x x) 3 1
1
16. 8x 3 (2x 5) 3 3x 2 (2x 5) 4
2x x 2 (2x 5) 3 (14x 15)
In Exercises 1–8, use the given substitution and the Chain Rule to
find dydx.
3 cos (3x 1)
5 cos (7 5x)
1. y sin (3x 1), u 3x 1 2. y sin (7 5x), u 7 5x
3. y cos (3x), u 3x 4. y tan (2x x 3 ), u 2x x 3
)
5. y ( sin x
1 cos x
2
sin x 2sin
x
, u 1 cos x (1 cos
x) 2
6. y 5 cot ( 2 x ) , u 2 7. y cos (sin x), u sin x
x
sin (sin x) cos x
8. y sec (tan x), u tan x
sec (tan x) tan (tan x) sec 2 x
In Exercises 9–12, an object moves along the x-axis so that its position
at any time t 0 is given by x(t) s(t). Find the velocity of the
object as a function of t.
4t sin ( 4t) cos ( 4t)
9. s cos ( p 2 3t ) 10. s t cos p 4t
4
11. s 3 p sin 3t 4
cos 5t
5p
12. s sin ( 3 p t
2 ) ( cos 7 p t
4 )
3 2
4 4 cos 3t sin 5t
3t
cos 7 7t
sin
2 4 4
In Exercises 13–24, find dydx. If you are unsure of your answer, use
csc x
NDER to support your computation.
csc x cot x
13. y x x 2 14. y csc x cot x 1
15. y sin 5 x cos 3 x 16. y x 3 2x 5 4
17. y sin 3 x tan 4x 18. y 4secx tan x
3
x
19. y 3(2x 1) 3/2 20. y (1 x 2 ) 3/2
2x 1
1 x
2
21. y sin 2 3x 2 22. y 1 cos 2x 2
In Exercises 33–38, find the value of f g at the given value of x.
33. f u u 5 1, u gx x, x 1 5/2
34. f u 1 1 u , u gx 1
, x 1 1
1 x
35. f u cot p u
, u gx 5x, x 1 4
10
1
36. f u u , cos 2 u gx px, x 1 u
4 5
2u
37. f u u
2
, u gx 10x
1
2 x 1, x 0 0
)
38. f u
( u 1
u 1
2
, u gx x
12
1, x 1 8
What happens if you can write a function as a composite in different
ways? Do you get the same derivative each time? The Chain Rule
says you should. Try it with the functions in Exercises 39 and 40.
39. Find dydx if y cos 6x 2 by writing y as a composite
with
(a) y cos u and u 6x 2. 6 sin (6x 2)
(b) y cos 2u and u 3x 1. 6 sin (6x 2)
40. Find dydx if y sin x 2 1 by writing y as a composite
with
(a) y sin u 1 and u x 2 . 2x cos (x 2 1)
(b) y sin u and u x 2 1. 2x cos (x 2 1)
In Exercises 41–48, find the equation of the line tangent to the curve
at the point defined by the given value of t.
41. x 2 cos t, y 2 sin t, t p4 y x 22
23. y 1 cos 2 7x 3 24. y tan 5x 5 2 (tan 5x)1/2 sec 2 5x
42. x sin 2pt, y cos 2pt, t 16 y 3x 2
In Exercises 25–28 find drdu.
43. x sec 2 t 1, y tan t, t p4 y 1
25. r tan 2 u sec 2 (2 ) 26. r sec 2u tan 2u
2 x 1 2
44. x sec t, y tan t, t p6 y 2x 3
27. r u sin u c os sin
28. r 2usecu sec ( tan 2)
2
sin
45. x t, y t, t 14 y x 1 4
In Exercises 29–32, find y.
46. x 2t 2 3, y t 4 , t 1 y x 4
29. y tan x 2 sec 2 x tan x 30. y cot x 2 csc 2 x cot x
47. x t sin t, y 1 cos t, t p3 y 3x 2
3
31. y cot 3x 1 32. y 9 tan x3 2 sec 2 x x
tan 3 48. x cos t, y 1 sin t, t p2 y 2
3
15. 5 sin 6 x cos x 3 cos 2 x sin x 18. 2 sec x sec x tan x
17. 4 sin 3 x sec 2 4x 3 sin 2 x cos x tan 4x 21. 6 sin (3x 2) cos (3x 2) 3 sin (6x 4)
22. 4 (1 cos 2x) sin 2x 26. 2 sec 3 2 2 sec 2 tan 2 2
23. 42(1 cos 2 7x) 2 cos 7x sin 7x 31. 18 csc 2 (3x 1) cot (3x 1)