# Second Derivatives Using Implicit Differentiation

According to my textbook, the second derivative of $y^{2}+xy-x^{2}=9$ is $\frac{90}{(2y+x)^{3}}$. The problem states "Express $\frac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$." I've tried for two days straight now, and I can't get that answer. I am convinced the book has a typo.

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What is the question? or what would you like to know? – anon Sep 18 '10 at 15:13
muad: "express $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}$ in terms of $x$ and $y$", per the OP. In other words, what is $y^{\prime\prime}(x)$ in terms of $x$ and $y$. – J. M. Sep 18 '10 at 16:06

## 2 Answers

Based on your work as linked in the comments on J.M.'s answer, you've very nearly got it (except for a typo in differentiating *: there's a dy/dx where there should be a d/dx, but the mathematics that follows is correct as if it were d/dx). The numerator you have is $$\begin{align} -2 ((2 x - y)^2 &+ (2 x - y) (2 y + x) - (2 y + x)^2) \\ &=-10x^2+10xy+10y^2 \\ &=10(y^2+xy-x^2) \\ &=10\cdot9 \\ &=90. \end{align}$$

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One little question: How did you get from the first step to the second step? When I factor it out, I get $2(5y^2+4xy-5x^2)$ – G.P. Burdell Sep 18 '10 at 19:50
I threw it into Mathematica and applied `Expand[]`.  Ignoring the -2 outside, the first item is $(2x-y)^2=4x^2-4xy+y^2$, the second is $(2x-y)(2y+x)=3xy-2y^2+2x^2$, and the third is $-(2y+x)^2=-4y^2-4xy-x^2$.  All together, $(4+2-1)x^2+(-4+3-4)xy+(1-2-4)y^2=5(x^2-xy-y^2)$. – Isaac Sep 18 '10 at 20:02
I hate basic errors. Thanks Isaac! – G.P. Burdell Sep 18 '10 at 20:09

Hint: treat $y$ as a function $y(x)$, so differentiating $xy(x)$ should give something like $x y^{\prime}(x)+y(x)$. Differentiate expressions twice, and solve for $y^{\prime\prime}(x)$

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I've tried that but I get a different result than the answer. I'll upload my work, just gimme an hour or so in LaTeX – G.P. Burdell Sep 18 '10 at 16:37
Hmm, if you didn't get $\frac{10(y(x+y)-x^2)}{(2y+x)^3}$ ... something has gone horribly wrong. – J. M. Sep 18 '10 at 16:53
– G.P. Burdell Sep 18 '10 at 18:06

1
M.: the numerator in your comment is $10(y(x+y)-x^2)=10(y^2+xy-x^2)=90$. – Isaac Sep 18 '10 at 18:32
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