Second Derivatives Using Implicit Differentiation

Question

According to my textbook, the second derivative of

\begin{equation*} y^{2}+xy-x^{2}=9 \end{equation*}

is

\begin{equation*} \frac{90}{(2y+x)^{3}}. \end{equation*}

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.

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. ain't a mathematician, Sep 18 '10 at 16:06

score 3 · Accepted Answer · edited Apr 13 '17 at 12:19

3

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 $$\begin{align} -2 ((2 x - y)^2 &+ (2 x - y) (2 y + x) - (2 y + x)^2) \\\\ &=-10x^2+10xy+10y^2 \\\\ &=10(y^2+xy-x^2) \\\\ &=10\cdot9 \\\\ &=90. \end{align}$$

edited Apr 13 '17 at 12:19

Community

1

answered Sep 18 '10 at 18:15

Isaac

36,557

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

score 1 · Answer 2 · answered Sep 18 '10 at 15:07

1

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 $y^{\prime\prime}(x)$

answered Sep 18 '10 at 15:07

J. M. ain't a mathematician

75,051

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. ain't a mathematician Sep 18 '10 at 16:53
Ok, so here's my work: http://imgur.com/4DDa7.png I'm still not getting what's supposed to be the right answer. – 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

Second Derivatives Using Implicit Differentiation

2 Answers2