# Easy Relation Second Derivate Proof

• Mar 30th 2008, 06:59 PM
Jeavus
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. :|
• Mar 30th 2008, 07:28 PM
Mathnasium
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.