5128_Ch03_pp098-184
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162 Chapter 3 Derivatives<br />
2. y 1 2 9 , x2 y<br />
3 2 2 9 x2<br />
3<br />
EXAMPLE 6 Using the Rational Power Rule<br />
d d<br />
(a) x x d x d x<br />
12 1 1<br />
2 x12 <br />
2 x<br />
Notice that x is defined at x 0, but 12x is not.<br />
d<br />
(b) x d x<br />
23 2 3 2<br />
x13 <br />
3x<br />
13<br />
The original function is defined for all real numbers, but the derivative is undefined at<br />
x 0. Recall Figure 3.12, which showed that this function’s graph has a cusp at x 0.<br />
d<br />
(c) cos x d x<br />
15 1 5 cos d<br />
x65 • cos x d x<br />
1 5 cos x65 sin x<br />
5. y 1 2x 3, x 2 y 2 2x 3 x 2<br />
Quick Review 3.7 (For help, go to Section 1.2 and Appendix A.5.)<br />
1 5 sin xcos x65 Now try Exercise 33.<br />
xy2<br />
8. y x 2 <br />
y x<br />
In Exercises 1–5, sketch the curve defined by the equation and find<br />
two functions y 1 and y 2 whose graphs will combine to give the curve.<br />
1. x y 2 0 y 1 x, y 2 x 2. 4x 2 9y 2 36<br />
3. x 2 4y 2 x x<br />
0 y 1 , y2 4. x 2 y 2 9<br />
2 2<br />
5. x 2 y 2 2x 3<br />
y 1 9 , x 2 y 2 9 x 2<br />
In Exercises 6–8, solve for y in terms of y and x.<br />
6. x 2 y2xy 4x y y 4x y 2xy<br />
<br />
x 2<br />
Section 3.7 Exercises<br />
In Exercises 1–8, find dydx.<br />
4. y x (x y)2 or 1 3x<br />
2xy<br />
<br />
x2<br />
1<br />
1. x 2 y xy 2 6 2 xy<br />
y2<br />
2. x 3 y 3 18xy 6 y x2<br />
2xy<br />
x2<br />
y 2 <br />
6x<br />
3. y 2 x 1 1<br />
<br />
4. x<br />
x 1<br />
x <br />
<br />
y<br />
y(x 1) 2<br />
x y<br />
5. x tan y cos 2 y 6. x sin y sec y<br />
1 y<br />
7. x tan xy 0 See page 164. 8. x sin y xy <br />
x cos y<br />
In Exercises 9–12, find dydx and find the slope of the curve at the<br />
indicated point.<br />
dy x<br />
,23<br />
9. x 2 y 2 dx y<br />
13, (2, 3)<br />
dy x<br />
10. x 2 y 2 9, (0, 3)<br />
,0<br />
dx y<br />
dy x 1<br />
11. (x 1) 2 (y 1) 2 13, (3, 4) , 23<br />
dx y 1<br />
12. (x 2) 2 (y 3) 2 25, (1, 7) See page 164.<br />
In Exercises 13–16, find where the slope of the curve is defined.<br />
13. x 2 y xy 2 4 See page 164. 14. x cos y See page 164.<br />
15. x 3 y 3 xy See page 164. 16. x 2 4xy 4y 2 3x 6<br />
See page 164.<br />
In Exercises 17–26, find the lines that are (a) tangent and<br />
(b) normal to the curve at the given point.<br />
17. x 2 xy y 2 1, 2, 3 (a) y 7 4 x 1 2 (b) y 4 7 x 2 9<br />
<br />
7<br />
18. x 2 y 2 25, 3, 4 (a) y 3 4 x 2 5<br />
<br />
4<br />
19. x 2 y 2 9, 1, 3<br />
(b) y 4 3 x<br />
See page 164.<br />
7. ysin x x cos x xyy<br />
8. xy 2 y yx 2 y<br />
y y x cos<br />
x<br />
<br />
sin<br />
x x<br />
In Exercises 9 and 10, find an expression for the function using<br />
rational powers rather than radicals.<br />
9. xx 3 x x 32 x 56 10. x 3<br />
x <br />
2 x<br />
x<br />
3<br />
12 x 56<br />
x<br />
25. (a) y 2x 2 (b) y 2 1<br />
2 <br />
20. y 2 2x 4y 1 0, 2, 1 (a) y x 1 (b) y x 3<br />
21. 6x 2 3xy 2y 2 17y 6 0, 1, 0 See page 164.<br />
22. x 2 3xy 2y 2 5, 3, 2 (a) y = 2 (b) y 3<br />
23. 2xy p sin y 2p, 1, p2 See page 164.<br />
24. x sin 2y y cos 2x, p4, p2 (a) y 2x (b) y 1 2 x 5 <br />
8<br />
25. y 2 sin px y, 1, 0<br />
26. x 2 cos 2 y sin y 0, 0, p (a) y (b) x 0<br />
In Exercises 27–30, use implicit differentiation to find dydx and then<br />
d 2 ydx 2 .<br />
27. x 2 y 2 1 See page 164. 28. x 23 y 23 1 See page 164.<br />
29. y 2 x 2 2x See page 164. 30. y 2 2y 2x 1<br />
See page 164.<br />
In Exercises 31–42, find dydx.<br />
31. y x 94 (9/4)x 5/4 32. y x 35 (3/5)x 8/5<br />
33. y 3 x (1/3)x 2/3 34. y 4 x (1/4)x 3/4<br />
35. y 2x 5 12 (2x 5) 32 36. y 1 6x 23 4(1 – 6x) 1/3<br />
37. y xx 2 x<br />
1 38. y (x 2 1) 3/2<br />
x 2 (x 2 1) 1/2 (x 2 1) 1/2<br />
x <br />
2<br />
1<br />
39. y 1 x See page 164. 40. y 32x 12 1 13<br />
See page 164.<br />
41. y 3csc x 32 See page 164. 42. y sin x 5 54<br />
See page 164.