I have two functions $u$ and $\phi$ given. I am not sure what they depend on, but I think that it is a common variable $\tau$. So $u(\tau)$ and $\phi(\tau)$. Then $\dot u$ is the derivative of $u$ with respect to $\tau$.

The derivation of a problem works if
$$\frac{\dot u}{\dot \phi} = \frac{\mathrm du}{\mathrm d\phi}.$$

Is that legitimate? Within thermodynamics, where an equation of state like $f(u, \phi) = 0$ holds, I learned that I the reciprocal of a derivative is the derivative the other way around, allowing to “cancel chain rules”. I am not sure whether this would hold here as well.

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This is probably just a duplicate of [how to calculate $\frac{d\dot{x}}{dx}$](https://math.stackexchange.com/q/103853/14291)? I can just cancel the $\mathrm d\tau$?