11X1 T09 08 implicit differentiation (2010)

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  • 1. Differentiability
  • 2. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a
  • 3. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y x 1 1 1 x
  • 4. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y x 1 1 1 x not differentiable at x = 1
  • 5. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y x 1 1 1 x not differentiable at x = 1 lim f x 1 x1
  • 6. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y x 1 1 1 x not differentiable at x = 1 lim f x 1 lim f x 1 x1 x1
  • 7. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y y x2 y x 1 1 1 x x not differentiable at x = 1 lim f x 1 lim f x 1 x1 x1
  • 8. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y y x2 y x 1 1 1 x 1 x not differentiable at x = 1 lim f x 1 lim f x 1 x1 x1
  • 9. Differentiability A function is differentiable at a point if the curve is smooth continuous i.e. lim f x lim f x x a x a y y y x2 y x 1 1 1 x 1 x not differentiable at x = 1 differentiable at x = 1 lim f x 1 lim f x 1 x1 x1
  • 10. Implicit Differentiation
  • 11. Implicit Differentiation df df dy dx dy dx
  • 12. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2
  • 13. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 x y d d 2 dx dx
  • 14. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 x y d d 2 dx dx d 2 dx y dy y dx d 2 dy
  • 15. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 x y d d 2 dx dx dy 1 2y dx d 2 dx y dy y dx d 2 dy
  • 16. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 x y d d 2 dx dx dy 1 2y dx dy 1 dx 2 y d 2 dx y dy y dx d 2 dy
  • 17. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 (ii) dx x y d 2 3 x y d d 2 dx dx dy 1 2y dx dy 1 dx 2 y d 2 dx y dy y dx d 2 dy
  • 18. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 (ii) dx x y d 2 3 x y d d 2 x2 3 y 2 y3 2 x dx dx dy dy dx 1 2y dx dy 1 dx 2 y d 2 dx y dy y dx d 2 dy
  • 19. Implicit Differentiation df df dy dx dy dx e.g. (i) x y 2 (ii) dx x y d 2 3 x y d d 2 x2 3 y 2 y3 2 x dx dx dy dy dx 1 2y 2 2 dy dx 3x y 2 xy 3 dy 1 dx dx 2 y d 2 dx y dy y dx d 2 dy
  • 20. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
  • 21. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9
  • 22. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 dy 2x 2 y 0 dx
  • 23. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 dy 2x 2 y 0 dx dy 2 y 2 x dx
  • 24. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 dy 2x 2 y 0 dx dy 2 y 2 x dx dy x dx y
  • 25. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 dy 2x 2 y 0 dx dy 2 y 2 x dx dy x dx y dy at 1, 2 2 , dx 1 2 2
  • 26. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 dy 2x 2 y 0 dx dy 2 y 2 x dx dy x dx y dy at 1, 2 2 , dx 1 2 2 1 required slope 2 2
  • 27. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 1 dy y2 2 x 1 2x 2 y 0 2 2 dx dy 2 y 2 x dx dy x dx y dy at 1, 2 2 , dx 1 2 2 1 required slope 2 2
  • 28. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 1 dy y2 2 x 1 2x 2 y 0 2 2 dx 2 2y 8 x 1 dy 2 y 2 x dx dy x dx y dy at 1, 2 2 , dx 1 2 2 1 required slope 2 2
  • 29. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2 x2 y 2 9 1 dy y2 2 x 1 2x 2 y 0 2 2 dx 2 2y 8 x 1 dy 2 y 2 x dx x 2 2y 9 0 dy x dx y dy at 1, 2 2 , dx 1 2 2 1 required slope 2 2
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