It's known that $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ . Today in class we were asked to find the relation between $\frac{d^2y}{dx^2}$ and $dy/dx$ and $dt$ by our professor. He also warned us that $\frac{d^2y}{dx^2}\neq \frac{d^2y/dt^2}{d^2x/dt^2}$. (I understood it) we also got a relation but now he is asking whether there exists a general term for relation between $dy/dx,d^ny/dx^n,dt$ ?
I don't think there is (maybe I am wrong) . How do I confirm it?
You already have $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$ differentiate both sides with respect to $x$ and use the quotient rule to get $$\frac{d^2y}{dx^2} = \frac{\displaystyle \frac{dx}{dt}\frac{d}{dx}\left(\frac{dy}{dt}\right) - \frac{dy}{dt}\frac{d}{dx}\left(\frac{dx}{dt}\right)}{\displaystyle \left(\frac{dx}{dt}\right)^2}$$
And then make use of $\dfrac{d}{dx}\dfrac{dy}{dt} = \dfrac{d}{dt} \dfrac{dy}{dt} \dfrac{dt}{dx} = \dfrac{dt}{dx}\dfrac{d^2y}{dt^2}$ and similar with the other term. Then clean up a bit, some nice cancellation happens.
