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Physics books gloss over math. Sometimes it bothers me.

Given a separable function of time and position f = h(x)g(t) then

d / dt of [inte] h(x)g(t)dx = [inte] h(x) dg(t)/dt dx

Where the derivative in the second integral is a partial deriv.

Why the chain rule does not apply is glossed over by the statement that the definite integral of f(x,t) with respect to x results in a new function that is a function of t only. I believe that, but since x can be a function of t I can't see why the chain rule does not apply. Can someone please explain this to me?

To be more specific the book says:

d/dt of [inte] x * psi(x,t)dx = [inte] x * dpsi(x,t)/dt dx

where again the second integral has partial deriv. w.r.t. time t. The '*' above just means times here.