SECTION 3.5 Implicit Differentiation 215 3.5 EXERCISES 1-4 29. xy (2x 2y2 x, (0.). (cardioid) (a) Find y' by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y 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. 9x2 y2 1 2. 2x2x + xy 1 3. 1 2 4. 30. x2/y2/4, (-3/3, 1), (astroid) - =4 5-20 Find dy/dx by implicit differentiation. 5. x2-4xy + y2 4 6. 2x2+xy y2 = 2 7. x+x'y2y 5 8. x-xy2 y= 1 31. 2(x2y225(x2- y2), ( 3, 1), (lemniscate) x2 10. xe xy 9. = y2 + 1 x+ y 12. Cos(xy) 1 + sin y 11. y cos x x2 + y 14. e' sin x x + xy 13. Vx +y x+ y Vx2 +y 15. e xy 16. xy 32. y(y 4) x(x -5), (0,-2), (devil's curve) 17. tan (xy) x + xy 18. x sin y + y sin x 1 20. tan(x y) 19. sin(xy) cos(x + y) 1x2 21. If f(x) +x[f (x)]' = 10 and f(1) = 2, find f'(1). 22. If g(x) +x sin g(x) x, find g' (0). 33. (a) The curve with equationy 5x-x2 is called a kampyle of Eudoxus. Find an equation of the tam line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the t line on a common screen. (If your graphing devi- graph implicitly defined curves, then use that ca ity. If not, you can still graph this curve by grap upper and lower halves separately.) 23-24 Regard y as the independent variable and x as the depen- dent variable and use implicit differentiation to find dx/dy. 24. y sec x x tan y 23. xy2-x'y + 2xy 0 25-32 Use implicit differentiation to find an equation of the tangent line to the curve at the given point. (T/2, T/4) x +3x2 is calle 34. (a) The curve with equation y Tschirnhausen cubic. Find an equation of the line to this curve at the point (1, -2). (b) At what points does this curve have horizonta tangents? (c) Illustrate parts (a) and (b) by graphing the cu tangent lines on a common screen. 25. y sin 2x x cos 2y, (T, T) 2x 2y, 26. sin(x + y) 27. xxy y2= 1, (2, 1) (hyperbola) 28. x2xy +4y 12, (2, 1) (ellipse)

Question

I need help with question 15 in Section 3.5, page 215, of the James Stewart Calculus Eighth Edition textbook.

SECTION 3.5 Implicit Differentiation
215
3.5 EXERCISES
1-4
29. xy (2x 2y2 x, (0.). (cardioid)
(a) Find y' by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get y
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. 9x2 y2 1
2. 2x2x + xy 1
3. 1
2
4.
30. x2/y2/4, (-3/3, 1), (astroid)
- =4
5-20 Find dy/dx by implicit differentiation.
5. x2-4xy + y2 4
6. 2x2+xy y2 = 2
7. x+x'y2y 5
8. x-xy2 y= 1
31. 2(x2y225(x2- y2), ( 3, 1), (lemniscate)
x2
10. xe xy
9.
= y2 + 1
x+ y
12. Cos(xy) 1 + sin y
11. y cos x x2 + y
14. e' sin x x + xy
13. Vx +y x+ y
Vx2 +y
15. e xy
16. xy
32. y(y 4) x(x -5), (0,-2), (devil's curve)
17. tan (xy) x + xy
18. x sin y + y sin x 1
20. tan(x y)
19. sin(xy) cos(x + y)
1x2
21. If f(x) +x[f (x)]' = 10 and f(1) = 2, find f'(1).
22. If g(x) +x sin g(x) x, find g' (0).
33. (a) The curve with equationy 5x-x2 is called a
kampyle of Eudoxus. Find an equation of the tam
line to this curve at the point (1, 2).
(b) Illustrate part (a) by graphing the curve and the t
line on a common screen. (If your graphing devi-
graph implicitly defined curves, then use that ca
ity. If not, you can still graph this curve by grap
upper and lower halves separately.)
23-24 Regard y as the independent variable and x as the depen-
dent variable and use implicit differentiation to find dx/dy.
24. y sec x x tan y
23. xy2-x'y + 2xy 0
25-32 Use implicit differentiation to find an equation of the
tangent line to the curve at the given point.
(T/2, T/4)
x +3x2 is calle
34. (a) The curve with equation y
Tschirnhausen cubic. Find an equation of the
line to this curve at the point (1, -2).
(b) At what points does this curve have horizonta
tangents?
(c) Illustrate parts (a) and (b) by graphing the cu
tangent lines on a common screen.
25. y sin 2x x cos 2y,
(T, T)
2x 2y,
26. sin(x + y)
27. xxy y2= 1, (2, 1) (hyperbola)
28. x2xy +4y 12, (2, 1) (ellipse)

Image Transcription

SECTION 3.5 Implicit Differentiation 215 3.5 EXERCISES 1-4 29. xy (2x 2y2 x, (0.). (cardioid) (a) Find y' by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y 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. 9x2 y2 1 2. 2x2x + xy 1 3. 1 2 4. 30. x2/y2/4, (-3/3, 1), (astroid) - =4 5-20 Find dy/dx by implicit differentiation. 5. x2-4xy + y2 4 6. 2x2+xy y2 = 2 7. x+x'y2y 5 8. x-xy2 y= 1 31. 2(x2y225(x2- y2), ( 3, 1), (lemniscate) x2 10. xe xy 9. = y2 + 1 x+ y 12. Cos(xy) 1 + sin y 11. y cos x x2 + y 14. e' sin x x + xy 13. Vx +y x+ y Vx2 +y 15. e xy 16. xy 32. y(y 4) x(x -5), (0,-2), (devil's curve) 17. tan (xy) x + xy 18. x sin y + y sin x 1 20. tan(x y) 19. sin(xy) cos(x + y) 1x2 21. If f(x) +x[f (x)]' = 10 and f(1) = 2, find f'(1). 22. If g(x) +x sin g(x) x, find g' (0). 33. (a) The curve with equationy 5x-x2 is called a kampyle of Eudoxus. Find an equation of the tam line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the t line on a common screen. (If your graphing devi- graph implicitly defined curves, then use that ca ity. If not, you can still graph this curve by grap upper and lower halves separately.) 23-24 Regard y as the independent variable and x as the depen- dent variable and use implicit differentiation to find dx/dy. 24. y sec x x tan y 23. xy2-x'y + 2xy 0 25-32 Use implicit differentiation to find an equation of the tangent line to the curve at the given point. (T/2, T/4) x +3x2 is calle 34. (a) The curve with equation y Tschirnhausen cubic. Find an equation of the line to this curve at the point (1, -2). (b) At what points does this curve have horizonta tangents? (c) Illustrate parts (a) and (b) by graphing the cu tangent lines on a common screen. 25. y sin 2x x cos 2y, (T, T) 2x 2y, 26. sin(x + y) 27. xxy y2= 1, (2, 1) (hyperbola) 28. x2xy +4y 12, (2, 1) (ellipse)

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