# differentiating under the integral sign

Suppose I have a moving curve $\alpha:I\times[0,T] \to \mathbb{R}^n.$ Its length is $$\int_\alpha ds = \int_I |\alpha_x|dx.$$ If I want to find the time derivative of this, I guess I differentiate under the integral sign.

If I have a function like $\int_I{|\alpha_{ss}|ds}$, how do I find its time derivative? Can I just differentiate the $|\alpha_{ss}|$ part inside the integral or do I need to do something with $ds$ too?

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Any presumptions about $\alpha$? –  copper.hat May 31 '12 at 22:22
Nope. I guess take it as smooth as needed. –  soup Jun 2 '12 at 21:35