uS sec ఇపె 2.5 Implicit Differentiation 149 d (as can 2.5 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. Famous Curves In Exercises 33-36, find the slope of the tangent line to the graph at the given point. CONCEPT CHECK 1. Explicit and Implicit Functions Describe the difference between the explicit form of a function and an 33. Witch of Agnesi: 34. Cissoid: implicit equation. Give an example of each. (x2 4)y = 8 (4 x)y2 = x3 2. Implicit Differentiation the guidelines for implicit differentiation. Implicit Differentiation Explain when you have to implicit differentiation to find a derivative. 1 In your own words, state y y (2, 2) 2 3 use Chain Rule How is the Chain Rule applied when (2, 1) finding dy/dx implicitly? 3 2 2 -2 -1 Derivative In Exercises 5-20, find -2 Finding dy/dx by implicit differentiation. 36. Folium of Descartes: 35. Bifolium: 3y36xy 0 (x2 y2) = 4x2y 6. x2-y225 5. x+29 7. xy5 16 9. x3-y2 11 8. 2x3+3y3= 64 4 10. xyy2x -2 12. Vxy = xy + 1 7 3 (1, 1) 2 11. x'y-y x 14. xty 8xy+3xy2 =9 ry) 2 13. x-32y2xy2 12 2 -1 -2 X 16. (sin 4 3 2 1 TX +cOS 15. sinx +2 cos 2y 18. cot y -2 x --y 17. csC x x(1 + tan y) 1 20. x sec - y Famous Curves In Exercises 37-42, find an equation of the tangent line to the graph at the given point. To print an graph, go to Math Graphs.com. 19. y = sinxy and enlarged copy of the Finding Derivatives Explicitly In Exercises 31-24, (a) find two explicit functions by solving he cquntion fer y in terms of x, (b) sketch the grapt of the cquatken and label the parts given by orraspondisg srxplicit functions, (c) differentiadhe explicit fnctions, and (d) find dy/dx implicisy and show ihat the result is equivalent to that of park (c). Implicitly 38. Circle 37. Parabola as (x+ 2)2+(y-3)2= 37 (y-3)2= 4(x-5) 10 10 (4, 4) 22. 25x236y 300 21. x2+ y2 64 23. 16y2- 2= 16 (6, 1) 4x + 6y +9= 0 24. x2y2 + 14 -2+ 2 46 8 4 6 -4-2 Finding the Slope of a Graph In Exercises 25-32, find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. -4 -6 40. Astroid 39. Cruciform x2/3y2/35 25. xy = 6, (-6, - 1) 26. 3x3y = 6, (1, 2) x2y2- 9x2-4y2 - 0 x2-36 x2-49 27. y2 x 49' (7,0) 29. (x + y)3 = x3 + y, (-1, 1) 30. y6xy - 1, (2, 3) 28. 4y3 11 12 36 (6,0) 4 (8, 1) (-4, 2/3) + 2 46 Xx 12 31. tan(x+y) = x, -6-4 -2 (0, 0) -4 (a9) 32. x COS y = 1, -12+ 2, 3 4 2 1