What is $\frac{d}{dx}\left(\frac{dx}{dt}\right)$? This question was inspired by the Lagrange equation, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What happens if the partial derivatives are replaced by total derivatives, leading to a situation where a function's derivative with respect to one variable is differentiated by the original function?

Write $$\frac{d}{dx}\left(\frac{dx}{dt}\right)=\frac{d}{dx}\left(\frac{1}{\frac{dt}{dx}}\right)$$ and use the chain rule.  $$\frac{d}{dx}\left(\frac{dx}{dt}\right)=\left(\frac{-1}{(\frac{dt}{dx})^2}\right)\frac{d^2t}{dx^2}=-\left (\frac{dx}{dt}\right )^2\frac{d^2t}{dx^2}$$
