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

Section 3.5 Implicit Differentiation 215
3.5
Exercises
1–4
(a) Find y9 by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get y9
in terms of x.
(c) Check that your solutions to parts (a) and (b) are consistent
by substituting the expression for y into your solution for
part (a).
1. 9x 2 2 y 2 − 1 2. 2x 2 1 x 1 xy − 1
29. x 2 1 y 2 − s2x 2 1 2y 2 2 xd 2 , s0, 1 2d, (cardioid)
y
x
3. sx 1 sy − 1
4.
2
x 2 1 y − 4
30. x 2y3 1 y 2y3 − 4, s23s3, 1d, (astroid)
y
5–20 Find dyydx by implicit differentiation.
5. x 2 2 4xy 1 y 2 − 4 6. 2x 2 1 xy 2 y 2 − 2
0 8
x
7. x 4 1 x 2 y 2 1 y 3 − 5 8. x 3 2 xy 2 1 y 3 − 1
9. x 2
x 1 y − y 2 1 1
10. xe y − x 2 y
11. y cos x − x 2 1 y 2 12. cossxyd − 1 1 sin y
31. 2sx 2 1 y 2 d 2 − 25sx 2 2 y 2 d, (3, 1), (lemniscate)
y
13. sx 1 y − x 4 1 y 4
14. e y sin x − x 1 xy
0
x
15. e xyy − x 2 y 16. xy − sx 2 1 y 2
17. tan 21 sx 2 yd − x 1 xy 2 18. x sin y 1 y sin x − 1
19. sinsxyd − cossx 1 yd 20. tansx 2 yd − y
1 1 x 2
32. y 2 sy 2 2 4d − x 2 sx 2 2 5d, (0, 22), (devil’s curve)
y
x
21. If f sxd 1 x 2 f f sxdg 3 − 10 and f s1d − 2, find f 9s1d.
22. If tsxd 1 x sin tsxd − x 2 , find t9s0d.
23–24 Regard y as the independent variable and x as the dependent
variable and use implicit differentiation to find dxydy.
23. x 4 y 2 2 x 3 y 1 2xy 3 − 0
24. y sec x − x tan y
25–32 Use implicit differentiation to find an equation of the
tangent line to the curve at the given point.
25. y sin 2x − x cos 2y, sy2, y4d
26. sinsx 1 yd − 2x 2 2y, s, d
27. x 2 2 xy 2 y 2 − 1, s2, 1d (hyperbola)
28. x 2 1 2xy 1 4y 2 − 12, s2, 1d (ellipse)
;
;
33. (a) The curve with equation y 2 − 5x 4 2 x 2 is called a
kampyle of Eudoxus. Find an equation of the tangent
line to this curve at the point s1, 2d.
(b) Illustrate part (a) by graphing the curve and the tangent
line on a common screen. (If your graphing device will
graph implicitly defined curves, then use that capability.
If not, you can still graph this curve by graphing its
upper and lower halves separately.)
34. (a) The curve with equation y 2 − x 3 1 3x 2 is called the
Tschirnhausen cubic. Find an equation of the tangent
line to this curve at the point s1, 22d.
(b) At what points does this curve have horizontal
tangents?
(c) Illustrate parts (a) and (b) by graphing the curve and the
tangent lines on a common screen.
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