How to find the second derivative of $2x^3 + y^3 = 5$? I'm finding it difficult to find the second derivative of the following equation: $2x^3 + y^3 = 5$.
My answer is $\dfrac{-4xy^3 - 8x^4}{y^5}$, but the answer key says $\dfrac{-20x}{y^5}$.
My friends and I have the same answer, but I don't know where I went wrong.
Here is my work:
I got the first derivative as $\dfrac{-2x^2}{y^2}$.
I then used the quotient rule to get 
$-2 \left(\dfrac{2xy^2 - 2x^2y\frac{dy}{dx}}{y^4}\right)$
After factoring out $2y$ from the numerator, I got $-4 \left(\dfrac{xy - x^2\left(\frac{-2x}{y^2}\right)}{y^3}\right)$. 
After splitting the fraction into $\dfrac{x}{y^2} + \dfrac{2x^4}{y^5}$, I ended up getting
$\dfrac{-4x}{y^2}-\dfrac{-8x^4}{y^5}$.
Could somebody point out my mistake? This isn't homework by the way. I'm preparing for a placement test tomorrow, but I'm not really sure where I went wrong.
 A: Your answer only requires some re-arrangement:
$$
2x^3 + y^3 = 5
$$
multiply by $x$ on both sides to get:
$$ 
2x^4 + xy^3 = 5 x
$$
or:
$$
xy^3 = 5x - 2x^4 
$$
...(substitute this in your equation to get):
$$
\frac{-4(5x - 2x^4 )- 8x^4}{y^5}\to y''=\frac{-20x}{y^5}
$$
A: As Cameron Williams points out in the comments, your answer is actually correct - just plug in $y^3 = 5 - 2x^3$ to the numerator in your expression $\displaystyle \frac{-4x(y^3) - 8x^4}{y^5}$ and you will get the answer given in the key.
You should keep in mind when using an answer key that often in mathematics, two things can look completely different but actually be the same.  A common example of this is with trigonometric expressions, where using a few identities an expression can be transformed entirely.  At any rate, before you assume you made a mistake, you might think about proving your answer is actually wrong.  In this case, you might try assuming that
$$
\frac{-4xy^3 - 8x^4}{y^5} = \frac{-20x}{y^5}
$$
and try to get a contradiction; but you would eventually arrive at something certainly true, and thus you would expect your answer and the key's answer to be equivalent.
A: You actually got the same answer as the key. Notice in your answer if you substitute (5-2x^3) for y^3, and simplify, the answers match. 
