<s>[INST] Implicit differentiation question Given that $x^n + y^n = 1$, show that $$\frac{d^2y}{dx^2} = -\frac{(n-1)x^{n-2}}{y^{2n-1}}.$$
I found that $\displaystyle nx^{n-1}+ny^{n-1}\frac{dy}{dx} = 0$ so that $\displaystyle y'=\frac{-x^{n-1}}{y^{n-1}}$.
Then $$n(n-1)x^{n-2}+n(n-1)y^{n-2}\left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2}ny^{n-1} = 0.$$
Therefore $$y'' = \displaystyle \frac{-n(n-1)x^{n-2}-n(n-1)y^{n-2}(y')^2}{ny^{n-1}}.$$
Substituting the first derivative:
$$y'' = \displaystyle\frac{-n(n-1)x^{n-2}-n(n-1)y^{n-2}\left(\dfrac{-x^{n-1}}{y^{n-1}}\right)^2}{ny^{n-1}}.$$
I've been trying tons of different steps and can't establish which way to eliminate the $y$ terms from the numerator. Could someone offer a hint on how to proceed. Thanks.
 [/INST] $y'=-x^{n-1}y^{1-n}$. 
$y''=-(n-1)x^{n-2}y^{1-n}-x^{n-1}(1-n)y^{-n}y'$
$y''=-(n-1)x^{n-2}y^{1-n}+x^{n-1}(1-n)y^{-n}x^{n-1}y^{1-n}$
$y''=-(n-1)x^{n-2}y^ny^{1-2n}+x^{n-1}(1-n)x^{n-1}y^{1-2n}$
$y''=-(n-1)x^{n-2}(1-x^n)y^{1-2n}+x^{n-1}(1-n)x^{n-1}y^{1-2n}$
Should be easy from there. 
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