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Switching order of derivatives

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If x(y,t) is a function of y and t and derivatives of x with respect to by y and t exist and are continuous, then the order of differentiation doesn't matter. This is called Clairaut's Theorem. I am sure I am missing some of the conditions and requirements, but in your case we would have d^(2)x/(dydt) = d^(2)x/(dtdy). This comes from multivariable calculus, and those "d"s should be "∂"s (partial derivatives), but hopefully that is good enough for your purposes.
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It's physics/engineering, people assume things are nice enough to do that all the time.

In general, Clairaut's theorem allows you to do that in very general context, but it's not literally always allowed. Also, you need to be in flat space, but I think that was assumed.

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