# 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.

• Your answer is correct. Use the fact that $y^3 = 5 - 2x^3$ and you will get what the answer key has. – Cameron Williams Aug 13 '13 at 16:15
• Ah, I see. Thanks! – kullalok Aug 13 '13 at 16:50

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): $$\frac{-4(5x - 2x^4 )- 8x^4}{y^5}\to y''=\frac{-20x}{y^5}$$

• Added the $\mathrm{\LaTeX{}}$ formatting to your equations, please refer to this usage to get more understandable math writing. – alandella Aug 13 '13 at 16:35
• i dont know how to use the latex can you teach me – Suraj M S Aug 13 '13 at 16:37
• You can refer to this page, it will give you the basic usage/related links to write with this language. – alandella Aug 13 '13 at 16:40
• does $$Latex$$ work in mobiles . i use mobile to answer questions. – Suraj M S Aug 13 '13 at 17:05
• On this site there is an interface, MathJax, which is basically a portable complier, the code therefore can be written everywhere; you just need to be in this page. – alandella Aug 13 '13 at 17:09

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.

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.