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

[karkas]

Homework Statement

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

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

Homework Equations

Considering that $$\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$$ how does one prove this statement?

The Attempt at a Solution

I've tried substituting like this:

$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}= \frac{d}{dx}\left(\frac{1}{\frac{dx}{dy}}\right)= \frac{ d \left( \frac{1}{u} \right)}{du}\frac{du}{dx}=-\left(\frac{1}{\frac{dx}{dy}}\right)^2 \frac{d\left(\frac{1}{\frac{dx}{dy}}\right)}{dx}$$

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

 

[tiny-tim]

hi karkas!

use the chain rule to replace that d/dx at the end by a d/dy

 

[karkas]

How would that give me the needed second derivative of the inverse function? I can't see it :(

 

[tiny-tim]

hint: what is d/dy of 1/(dx/dy) ?

 

[karkas]

Its (-d^2x/dy^2)/(dx/dy)^2 isn't it?

 

[tiny-tim]

then doesn't that solve the question?

 

[karkas]

then doesn't that solve the question?

oh man I should really learn to improve my writing...