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Math Help - Easy Relation Second Derivate Proof

  1. #1
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    Easy Relation Second Derivate Proof

    I can't do this for the life of me, I should be able to. Argh.

    For the relation 3x^2 - y^2 = 7, show that y'' = -21/y^3.

    Do I use implicit differentiation after re-arranging for y?

    I'm stuck. :|
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  2. #2
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    Solving for y:

    y = (3x^2-7)^{\frac{1}{2}}

    Then, using the chain rule:

    y' = \frac{1}{2} (3x^2 - 7)^{- \frac{1}{2}}*(6x) = \frac {3x}{(3x^2-7)^{\frac{1}{2}}}

    Now, differentiating again, using the quotient rule:

    y'' = \frac{3(3x^2-7)^{\frac{1}{2}} - 3x* \frac{1}{2} (3x^2-7)^{- \frac{1}{2}}(6x)}{3x^2-7}

    which does work out to \frac{-21}{y^3}.

    Let me know if you have trouble with the simplifying, but I assume your question was more about the differentiation.
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