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)