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Introduction Derivatives Related Rates Inverse Functions Implicit Differentiation Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation Introduction Derivatives

  1. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  2. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  3. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 � y ( x ) 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  4. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x ) � y ( x ) 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  5. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  6. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  7. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  8. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  9. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x 2 x y ′ = 3 y 2 + 6 y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  10. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x 2 x y ′ = 3 y 2 + 6 y y ′ � � � ( 0 , 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  11. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x 2 x y ′ = 3 y 2 + 6 y � 2 x y ′ � � = � 3 y 2 + 6 y � � ( 0 , 1 ) � ( 0 , 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  12. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x 2 x y ′ = 3 y 2 + 6 y � 2 x y ′ � � = = 0 � 3 y 2 + 6 y � � ( 0 , 1 ) � ( 0 , 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  13. Introduction Derivatives Related Rates Inverse Functions Example. The function y satisfies y 3 + 3 y 2 − 4 = x 2 . Find its derivative at the point ( 0 , 1 ) . � 3 + 3 � 2 − 4 x 2 � � y ( x ) y ( x ) = d � 3 + 3 d �� � 2 − 4 � � x 2 � � y ( x ) y ( x ) = dx dx � 2 y ′ ( x )+ 6 � � � y ′ ( x ) y ( x ) y ( x ) = 3 2 x 3 y 2 y ′ + 6 yy ′ = 2 x y ′ � 3 y 2 + 6 y � = 2 x 2 x y ′ = 3 y 2 + 6 y � 2 x y ′ � � = = 0 � 3 y 2 + 6 y � � ( 0 , 1 ) � ( 0 , 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  14. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  15. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  16. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  17. Introduction Derivatives Related Rates Inverse Functions Algorithm. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  18. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  19. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  20. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  21. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  22. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  23. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  24. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  25. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. 2. Differentiate both sides of the equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  26. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. 2. Differentiate both sides of the equation. Remember to use the chain rule for terms with y. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  27. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. 2. Differentiate both sides of the equation. Remember to use the chain rule for terms with y. 3. Solve for y ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  28. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. 2. Differentiate both sides of the equation. Remember to use the chain rule for terms with y. 3. Solve for y ′ . This is always possible, because, in any term that contains y ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  29. Introduction Derivatives Related Rates Inverse Functions Algorithm. Computing a derivative by implicit differentiation . 1. In the equation, determine the independent variable, usually it’s x, and the dependent variable, that is, the function, which is usually y. To start with, it may be a good idea to replace “y” with “y ( x ) ” or with “f ( x ) ”. 2. Differentiate both sides of the equation. Remember to use the chain rule for terms with y. 3. Solve for y ′ . This is always possible, because, in any term that contains y ′ , the quantity y ′ is just a factor. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  30. Introduction Derivatives Related Rates Inverse Functions Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  31. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  32. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  33. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  34. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  35. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  36. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  37. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 d d � xy 3 + xy + 3 x � = dx ( 2 ) dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  38. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 d d � xy 3 + xy + 3 x � = dx ( 2 ) dx y 3 + x 3 y 2 y ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  39. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 d d � xy 3 + xy + 3 x � = dx ( 2 ) dx y 3 + x 3 y 2 y ′ + y + xy ′ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  40. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 d d � xy 3 + xy + 3 x � = dx ( 2 ) dx y 3 + x 3 y 2 y ′ + y + xy ′ + 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  41. Introduction Derivatives Related Rates Inverse Functions Example. Find the equation of the tangent line of the graph defined by the equation xy 3 + xy + 3 x = 2 at the point ( 2 , − 1 ) . Dependent and independent variables. Independent: x , dependent: y . Differentiation with respect to x. xy 3 + xy + 3 x = 2 d d � xy 3 + xy + 3 x � = dx ( 2 ) dx y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  42. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  43. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  44. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  45. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  46. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x y ′ � � � ( 2 , − 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  47. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  48. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = 3 · 2 · ( − 1 ) 2 + 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  49. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = − 1 = 3 · 2 · ( − 1 ) 2 + 2 8 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  50. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = − 1 = 3 · 2 · ( − 1 ) 2 + 2 8 Equation of the tangent line logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  51. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = − 1 = 3 · 2 · ( − 1 ) 2 + 2 8 Equation of the tangent line: y = − 1 8 ( x − 2 ) − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  52. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = − 1 = 3 · 2 · ( − 1 ) 2 + 2 8 Equation of the tangent line: y = − 1 8 ( x − 2 ) − 1 = − 1 8 x − 3 4. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  53. Introduction Derivatives Related Rates Inverse Functions Solving for y ′ . y 3 + x 3 y 2 y ′ + y + xy ′ + 3 = 0 � 3 xy 2 + x � � y 3 + y + 3 � y ′ = − − y 3 + y + 3 y ′ = 3 xy 2 + x − y 3 + y + 3 � y ′ � � = � 3 xy 2 + x � � ( 2 , − 1 ) � ( 2 , − 1 ) − ( − 1 ) 3 +( − 1 )+ 3 = − 1 = 3 · 2 · ( − 1 ) 2 + 2 8 Equation of the tangent line: y = − 1 8 ( x − 2 ) − 1 = − 1 8 x − 3 4. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  54. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  55. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  56. Introduction Derivatives Related Rates Inverse Functions Graphical Check logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  57. Introduction Derivatives Related Rates Inverse Functions Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  58. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  59. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  60. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  61. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  62. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  63. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  64. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 h . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  65. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 h . Sides rise at a 45 ◦ angle logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  66. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 h . Sides rise at a 45 ◦ angle ⇒ r = h . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  67. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 h . Sides rise at a 45 ◦ angle ⇒ r = h . So the volume of this cone, in terms of its height, is logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  68. Introduction Derivatives Related Rates Inverse Functions Example. Gravel that is dumped off a conveyor belt forms a right circular cone whose sides rise at a 45 ◦ angle. If 2 m 3 of gravel are dumped onto the cone every second, how fast is the height of the cone rising when the cone is 10 m tall? Volume of a right circular cone: V = 1 3 π r 2 h . Sides rise at a 45 ◦ angle ⇒ r = h . So the volume of this cone, in terms of its height, is V ( h ) = 1 3 π h 3 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  69. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  70. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  71. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ V ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  72. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  73. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  74. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 h ′ � = � � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  75. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 V ′ � h ′ � � = � � π h 2 � h = 10 m � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  76. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 2 m 3 V ′ � h ′ � � s = = � � π h 2 π ( 10 m ) 2 � h = 10 m � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  77. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 2 m 3 V ′ � 1 m h ′ � � s = = π ( 10 m ) 2 = � � π h 2 50 π s � h = 10 m � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  78. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 2 m 3 V ′ � 1 m s ≈ 0 . 0064 m h ′ � � s = = π ( 10 m ) 2 = � � π h 2 50 π s � h = 10 m � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  79. Introduction Derivatives Related Rates Inverse Functions 1 3 π h 3 = V � 1 � d d 3 π h 3 = dtV dt 1 3 π 3 h 2 h ′ = π h 2 h ′ V ′ = V ′ h ′ = π h 2 2 m 3 V ′ � 1 m s ≈ 0 . 0064 m h ′ � � s = = π ( 10 m ) 2 = � � π h 2 50 π s � h = 10 m � h = 10 m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  80. Introduction Derivatives Related Rates Inverse Functions Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  81. Introduction Derivatives Related Rates Inverse Functions Example. Compute the derivative of f ( x ) = arctan ( x ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  82. Introduction Derivatives Related Rates Inverse Functions Example. Compute the derivative of f ( x ) = arctan ( x ) . y = arctan ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

  83. Introduction Derivatives Related Rates Inverse Functions Example. Compute the derivative of f ( x ) = arctan ( x ) . y = arctan ( x ) tan ( y ) = x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implicit Differentiation

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