11 x1 t09 08 implicit differentiation (2013)

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11 x1 t09 08 implicit differentiation (2013)

  1. 1. Differentiability
  2. 2. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   
  3. 3. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x 
  4. 4. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x not differentiable at x = 1
  5. 5. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x  1lim 1xf x  not differentiable at x = 1
  6. 6. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x  1lim 1xf x    1lim 1xf x not differentiable at x = 1
  7. 7. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x  1lim 1xf x    1lim 1xf x not differentiable at x = 1yx2y x
  8. 8. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x  1lim 1xf x    1lim 1xf x not differentiable at x = 1yx2y x1
  9. 9. DifferentiabilityA function is differentiable at a point if the curve is smooth continuous   i.e. lim limx a x af x f x   yx111y x  1lim 1xf x    1lim 1xf x not differentiable at x = 1yx2y x1differentiable at x = 1
  10. 10. Implicit Differentiation
  11. 11. Implicit Differentiationdf df dydx dy dx 
  12. 12. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y
  13. 13. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y   2d dx ydx dx
  14. 14. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y   2d dx ydx dx   2 2d d dyy ydx dy dx 
  15. 15. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y1 2dyydx   2d dx ydx dx   2 2d d dyy ydx dy dx 
  16. 16. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y1 2dyydx   2d dx ydx dx12dydx y   2 2d d dyy ydx dy dx 
  17. 17. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y1 2dyydx   2d dx ydx dx12dydx y   2 2d d dyy ydx dy dx  2 3(ii)dx ydx
  18. 18. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y1 2dyydx   2d dx ydx dx12dydx y   2 2d d dyy ydx dy dx  2 3(ii)dx ydx    2 2 33 2dyx y y xdx    
  19. 19. Implicit Differentiationdf df dydx dy dx 2e.g. (i) x y1 2dyydx   2d dx ydx dx12dydx y   2 2d d dyy ydx dy dx  2 3(ii)dx ydx    2 2 33 2dyx y y xdx    2 2 33 2dyx y xydx 
  20. 20.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 
  21. 21.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 
  22. 22.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 
  23. 23.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx 
  24. 24.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y 
  25. 25.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 
  26. 26.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 1required slope2 2  
  27. 27.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 1required slope2 2   12 2 12 2y x   
  28. 28.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 1required slope2 2   12 2 12 2y x   2 2 8 1y x   
  29. 29.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 1required slope2 2   12 2 12 2y x   2 2 8 1y x   2 2 9 0x y  
  30. 30.    2 2Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 29x y 2 2 0dyx ydx 2 2dyy xdx dy xdx y   1at 1,2 2 ,2 2dydx 1required slope2 2   12 2 12 2y x   2 2 8 1y x   2 2 9 0x y  Exercise 7K; 1acegi, 2bdfh, 3a,4a, 7, 8

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