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The usage of chain rule in physics

I often see in physics that, we say that we can multiply infinitesimals to use chain rule. For example,

$$ \frac{dv}{dt} = \frac{dv}{dx} \cdot v(t)$$

But, what bothers me about this is that it raises some serious existence questions for me; when we say that we take the derivative of $v$ velocity with respect to distance, that means we can write velocity as a function of distance. But, how do we know that this is always possible? As in, when we do these multiplications of differentials we are implicitly assuming that $v$ can be changed from a function of time into a function of displacement.

I see this used ubiquitously, and there are some crazier variations I've seen of literally swapping differentials like $ dv \frac{dm}{dt} = dm \frac{dv}{dt}$ , as shown by the answer of user "Fakemod" in this stack post.

Answer

I like this question and there already some good answers.  I'm not going to repeat those, but I wanted to add a couple of points focused on the second part of your question regarding "swapping" differentials.

The first is that the presence of a differential quantity is an abstraction that is usually only useful as an intermediate step in calculating something else.  By that, I mean you never *measure* something like $\rho\ dV$ directly.  You can only hope to measure:
1. The integral of that quantity $\int \rho\ dV$ over some volume (equivalently, you back off of the abstraction and measure $\rho \Delta V$ for some finite volume $\Delta V$)  --OR--
2. The "ratio of differentials" (being deliberately loose for the moment), which in the limit is a derivative.  So an expression like $f(t) dt = g(x) dx$ gets "divided through" to be $f(t) = g(x) (dx/dt) = g(x)v(t)$.  We believe we know how to measure changes in quantities and gradients.

That's relevant to the second part of your question about "swapping" differentials because when that's done legitimately, it typically works because you're ultimately going to put that expression under an integral sign, and the notation conveniently reflects (some might prefer to say the notation is easily abused when applying) the integration-by-substitution rule
$$ \int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$$
which you could rewrite in Leibnitz notation for $u = g(x)$ and get the appearance that you are swapping or canceling differentials.

Since the integration-by-substitution rule is basically the chain rule in reverse, however, all of this begs your initial question of why the chain rule is valid in physics.  For that, I refer back to the other already-good answers.

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