Integrate time dilation with derivative

[Franyer]

I need to integrate time dilation with derivative, how could I do that?

 

[Nugatory]

I need to integrate time dilation with derivative, how could I do that?

Without knowing the problem you're trying to analyze it's hard to answer with any certainty... But the chances are very good that you will be better off abandoning the time dilation formula and working with the more general Lorentz transformations.
The time dilation formula is a special case of the Lorentz transformations, and applicable only under very limited circumstances.

But tell us more about what you're trying to do and you'll get better and more helpful answer.