<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] Expanding your final expression, we have 
\begin{align*}
y'' &= \frac{-n(n-1)x^{n-2} -n(n-1)y^{n-2}\dfrac{x^{2n-2}}{y^{2n-2}}}{ny^{n-1}}\\
&= \frac{-n(n-1)x^{n-2} -n(n-1)y^{-n}x^{2n-2}}{ny^{n-1}}\\
&= \frac{-n(n-1)x^{n-2}(1 + x^{n}y^{-n})}{ny^{n-1}}\\
&= \frac{-(n-1)x^{n-2}(1 + x^{n}y^{-n})}{y^{n-1}}.
\end{align*}
Now note that $x^n + y^n = 1$, so $x^ny^{-n} + 1 = y^{-n}$. Therefore
\begin{align*}
y'' &= \frac{-(n-1)x^{n-2}(1 + x^{n}y^{-n})}{y^{n-1}}\\
&= \frac{-(n-1)x^{n-2}y^{-n}}{y^{n-1}}\\
&= \frac{-(n-1)x^{n-2}}{y^{2n-1}}.
\end{align*}
</s>