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Math Help - Find y'' by implicit differentation

  1. #1
    DBA
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    Find y'' by implicit differentation

    I have
    9x^2 +y^2 = 9

    I did

    18x + 2yy' = 0

    2yy' = -18x

    y' = -18x/2y = -9x/y

    where y'= dy/dx = d/dx depending what notation you use.

    y'' = d/dx[ -9x/y]

    = -9x * (-1y^-2)*y' + y^-1 *(-9)

    = (9xy' / y^2 ) - (9/y)

    Then I plug in y'

    y'' = (9x * (-9xy^-1) / y^2 ) - (9/y)

    y'' = -9* ((9x^2 + y^2)/(y^3)) this is right according to the solution

    But the solution goes on with the next step:

    y'' = -9 * (9/y^3) = -81/y^3

    And I do not know how they get this last step.

    Can someone help with that?

    Thanks
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  2. #2
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    Quote Originally Posted by DBA View Post
    I have
    9x^2 +y^2 = 9

    I did

    18x + 2yy' = 0

    2yy' = -18x

    y' = -18x/2y = -9x/y

    where y'= dy/dx = d/dx depending what notation you use.

    y'' = d/dx[ -9x/y]

    = -9x * (-1y^-2)*y' + y^-1 *(-9)

    = (9xy' / y^2 ) - (9/y)

    Then I plug in y'

    y'' = (9x * (-9xy^-1) / y^2 ) - (9/y)

    y'' = -9* ((9x^2 + y^2)/(y^3)) this is right according to the solution

    9x^2+y^2 = 9 from the original equation

    But the solution goes on with the next step:

    y'' = -9 * (9/y^3) = -81/y^3

    And I do not know how they get this last step.

    Can someone help with that?

    Thanks
    ...
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  3. #3
    DBA
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    Can you explain what you did, I do not understand.

    I want to know how they got y" = -81/y^3
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  4. #4
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    Quote Originally Posted by DBA View Post
    Can you explain what you did, I do not understand.

    I want to know how they got y" = -81/y^3
    y'' = -9* ((9x^2 + y^2)/(y^3))

    the red expression is 9

    y'' = -9*(9)/(y^3) = -81/y^3
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  5. #5
    DBA
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    Oh, I did not see that. Thanks you
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