5128_Ch03_pp098-184
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Section 3.6 Chain Rule 153<br />
Quick Review 3.6<br />
(For help, go to Sections 1.2 and 1.6.)<br />
In Exercises 1–5, let f x sin x, gx x 2 1, and hx 7x.<br />
Write a simplified expression for the composite function.<br />
1. f gx sin (x 2 1) 2. f ghx sin (49x 2 1)<br />
3. g hx 49x 2 1 4. h gx 7x 2 7<br />
5. f ( g x<br />
hx<br />
)<br />
sin x2 7<br />
1<br />
<br />
x<br />
In Exercises 6–10, let f x cos x, gx x 2, and<br />
hx 3x 2 . Write the given function as a composite of two or more<br />
of f, g, and h. For example, cos 3x 2 is f hx.<br />
6. cosx 2 g(f(x)) 7. 3 cos 2 x 2 g(h(f(x)))<br />
8. 3 cos x 6 h(g(f(x))) 9. cos 27x 4 f(h(h(x)))<br />
10. cos 2 3x 2 f(g(h(x)))<br />
3. 3 sin (3x)<br />
4. (2 3x 2 ) sec 2 (2x – x 3 )<br />
Section 3.6 Exercises<br />
6. 1 0<br />
x<br />
2 csc 2 2 x 9. 3 sin 2 3t 13. 2(x x) 3 1 <br />
1<br />
16. 8x 3 (2x 5) 3 3x 2 (2x 5) 4<br />
2x x 2 (2x 5) 3 (14x 15)<br />
In Exercises 1–8, use the given substitution and the Chain Rule to<br />
find dydx.<br />
3 cos (3x 1)<br />
5 cos (7 5x)<br />
1. y sin (3x 1), u 3x 1 2. y sin (7 5x), u 7 5x<br />
3. y cos (3x), u 3x 4. y tan (2x x 3 ), u 2x x 3<br />
)<br />
5. y ( sin x<br />
1 cos x<br />
2<br />
sin x 2sin<br />
x<br />
, u 1 cos x (1 cos<br />
x) 2<br />
6. y 5 cot ( 2 x ) , u 2 7. y cos (sin x), u sin x<br />
x<br />
sin (sin x) cos x<br />
8. y sec (tan x), u tan x<br />
sec (tan x) tan (tan x) sec 2 x<br />
In Exercises 9–12, an object moves along the x-axis so that its position<br />
at any time t 0 is given by x(t) s(t). Find the velocity of the<br />
object as a function of t.<br />
4t sin ( 4t) cos ( 4t)<br />
9. s cos ( p 2 3t ) 10. s t cos p 4t<br />
4<br />
11. s 3 p sin 3t 4<br />
cos 5t<br />
5p <br />
12. s sin ( 3 p t<br />
2 ) ( cos 7 p t<br />
4 )<br />
3 2<br />
4 4 cos 3t sin 5t<br />
<br />
3t<br />
cos 7 7t<br />
sin <br />
2 4 4<br />
In Exercises 13–24, find dydx. If you are unsure of your answer, use<br />
csc x<br />
NDER to support your computation. <br />
csc x cot x<br />
13. y x x 2 14. y csc x cot x 1<br />
15. y sin 5 x cos 3 x 16. y x 3 2x 5 4<br />
17. y sin 3 x tan 4x 18. y 4secx tan x<br />
3<br />
x<br />
19. y 3(2x 1) 3/2 20. y (1 x 2 ) 3/2<br />
2x 1<br />
1 x <br />
2<br />
21. y sin 2 3x 2 22. y 1 cos 2x 2<br />
In Exercises 33–38, find the value of f g at the given value of x.<br />
33. f u u 5 1, u gx x, x 1 5/2<br />
34. f u 1 1 u , u gx 1<br />
, x 1 1<br />
1 x<br />
35. f u cot p u<br />
, u gx 5x, x 1 4<br />
10<br />
1<br />
36. f u u , cos 2 u gx px, x 1 u<br />
4 5<br />
2u<br />
37. f u u<br />
2<br />
, u gx 10x<br />
1<br />
2 x 1, x 0 0<br />
)<br />
38. f u <br />
( u 1<br />
u 1<br />
2<br />
, u gx x<br />
12<br />
1, x 1 8<br />
What happens if you can write a function as a composite in different<br />
ways? Do you get the same derivative each time? The Chain Rule<br />
says you should. Try it with the functions in Exercises 39 and 40.<br />
39. Find dydx if y cos 6x 2 by writing y as a composite<br />
with<br />
(a) y cos u and u 6x 2. 6 sin (6x 2)<br />
(b) y cos 2u and u 3x 1. 6 sin (6x 2)<br />
40. Find dydx if y sin x 2 1 by writing y as a composite<br />
with<br />
(a) y sin u 1 and u x 2 . 2x cos (x 2 1)<br />
(b) y sin u and u x 2 1. 2x cos (x 2 1)<br />
In Exercises 41–48, find the equation of the line tangent to the curve<br />
at the point defined by the given value of t.<br />
41. x 2 cos t, y 2 sin t, t p4 y x 22<br />
23. y 1 cos 2 7x 3 24. y tan 5x 5 2 (tan 5x)1/2 sec 2 5x<br />
42. x sin 2pt, y cos 2pt, t 16 y 3x 2<br />
In Exercises 25–28 find drdu.<br />
43. x sec 2 t 1, y tan t, t p4 y 1<br />
25. r tan 2 u sec 2 (2 ) 26. r sec 2u tan 2u<br />
2 x 1 2 <br />
44. x sec t, y tan t, t p6 y 2x 3<br />
27. r u sin u c os sin <br />
28. r 2usecu sec ( tan 2)<br />
2<br />
sin<br />
45. x t, y t, t 14 y x 1 4 <br />
In Exercises 29–32, find y.<br />
46. x 2t 2 3, y t 4 , t 1 y x 4<br />
<br />
29. y tan x 2 sec 2 x tan x 30. y cot x 2 csc 2 x cot x<br />
47. x t sin t, y 1 cos t, t p3 y 3x 2 <br />
3<br />
31. y cot 3x 1 32. y 9 tan x3 2 sec 2 x x<br />
tan 3 48. x cos t, y 1 sin t, t p2 y 2<br />
3<br />
15. 5 sin 6 x cos x 3 cos 2 x sin x 18. 2 sec x sec x tan x<br />
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)<br />
22. 4 (1 cos 2x) sin 2x 26. 2 sec 3 2 2 sec 2 tan 2 2<br />
23. 42(1 cos 2 7x) 2 cos 7x sin 7x 31. 18 csc 2 (3x 1) cot (3x 1)