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?

Alternatively, the chain rule $\frac{dy}{dx} = \frac{dy}{dt}\frac{dt}{dx}$ gives $$\frac{d}{dx}(\frac{dx}{dt}) = \frac{d}{dt}(\frac{dx}{dt})\frac{dt}{dx} = \frac{d^2x}{dt^2}\frac{dt}{dx}.$$
Of course, this is the same as Ross' answer above.  That is, we have the identity $$-\left(\frac{dx}{dt}\right)^2\frac{d^2t}{dx^2} = \frac{d^2x}{dt^2}\frac{dt}{dx},$$
which follows from differentiating the equation $\frac{dx}{dt}\frac{dt}{dx} = 1$ with respect to $t$.
