Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 3 down vote favorite

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.

share|improve this question
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

up vote 3 down vote accepted

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

share|improve this answer
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
up vote 1 down vote

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

share|improve this answer
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
Ok, so here's my work: 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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.