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How could I prove that the identity $ \frac{\partial y}{\partial x} \cdot\frac{\partial x}{\partial y} = 1 $ is equivalent to the relation

$ (\frac{\partial x}{\partial y})^2 \frac{\partial^3 y}{\partial x^3}+3\frac{\partial^2 y}{\partial x^2} \frac{\partial^2 x}{\partial y^2} + (\frac{\partial y}{\partial x})^2 \frac{\partial^3 x}{\partial y^3} = 0$ ?

I don´t know how to proceed since I don´t have very clear the difference between an identity and a relation. Any help would be very appreciated.

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    I think you want to flip one of the derivatives in your identity. Try starting by differentiating the identity w.r.t $x$ twice and w.r.t $y$ twice – 2017-02-08

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The calculus below shows how to derive Eq.(2) from Eq.(1)

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