(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In one of my class's tests I've come across the following equation:

[tex]\frac{d^2 y}{dx^2} \: + \: \left(\frac{dy}{dx}\right)^3 \frac{d^2 x}{dy^2} \: =0[/tex]

2. Relevant equations

Considering that [tex]\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}[/tex] how does one prove this statement?

3. The attempt at a solution

I've tried substituting like this:

[tex]\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}= \frac{d}{dx}\left(\frac{1}{\frac{dx}{dy}}\right)= \frac{ d \left( \frac{1}{u} \right)}{du}\frac{du}{dx}=-\left(\frac{1}{\frac{dx}{dy}}\right)^2 \frac{d\left(\frac{1}{\frac{dx}{dy}}\right)}{dx}[/tex]

but i don't see how this can continue to finalize the proof...

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# Proving an equation of f and f^(-1) derivatives

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