Are these same equation

Question

How is $\frac{d}{dy}(\frac{dy}{dx}) = \frac{dx}{dy}\frac{d}{dx}(\frac{dy}{dx})$

I was looking at the answer to this question Prove $\frac{d^2y}{dx^2}=-\frac{\frac{d^2x}{dy^2}}{(\frac{dx}{dy})^3}$

I believe this is application of chain rule, but could not intuitively get this. Can I write the above equation as
$\frac{d}{dy}(\frac{dy}{dx}) = \frac{d}{dx}(\frac{dy}{dx})\frac{dx}{dy}$

Answer 1

According to chain rule, we have, using Leibniz notation

$$\frac{du}{dy}=\frac{dx}{dy}\frac{du}{dx}$$

If $u$ is a function of $y$ and $y$ a function of $x$.

Here, we have $u=\left(\frac{dy}{dx}\right)$, so

$$\frac d{dy}\left(\frac{dy}{dx}\right)=\frac{dx}{dy}\frac d{dx}\left(\frac{dy}{dx}\right)$$