Hi, Jim

Reynold's theorem, like Leibnitz integral theorem, which is the result of Leibnitz rule, stems from the differentiation of a product of two functions, u and v of the dependent variable(s).

This is why it is said that Leibnitz is the one dimensional version.

http://mathworld.wolfram.com/Reynold...rtTheorem.html
So I assume by one dimensional you mean you have only one independent variable and one dependent one. ie you have some funtion y = f(x) so you can write the expression in the form

$\displaystyle \frac{d}{{dx}}\left[ I \right] = \frac{d}{{dx}}\int_u^v {ydx} $

where u and v are functions of x alone.

Can you confirm if this is what you mean or give an example if it is not?

I can then supply the proof you seek which is based on the Pascal's triangle (or the binomial expansion) and induction.

By the way, why did you post in High school Physics?

This is hardly that although I would say that the above proof scheme would have been accessible of a UK student of the old fashioned A level.