I stumbled upon this on a "worked solutions manual" But I dont quite understand the part I selected in red. [![This is the working out they show][1]][1]
How can $ \frac{dy}{dt}$ be converted to $\frac{d^2y}{dx^2}$ when differentiating with respect to $x$?
Im probably missing something... [1]: https://i.stack.imgur.com/YZ3FC.jpg
You have $$\frac{dy}{dx}=\frac{1}{2x}\frac{dy}{dt},~\frac{dx}{dt}=2x$$ Then $$\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dt}(\frac{dy}{dx})(\frac{dt}{dx})=\frac{d}{dt} \left(\frac{1}{2x} \frac{dy}{dt}\right)\frac{1}{2x}$$ $$=\left(\frac{-1}{2x^2}\frac{dx}{dt}\frac{dy}{dt}+\frac{1}{2x}\frac{d^2y}{dt^2}\right)\frac{1}{2x}=\frac{1}{4x^2}\left(\frac{d^2y}{dt^2}-2\frac{dy}{dt}\right)$$
