Variable Limits and the Chain Rule

The Fundamental Theorem handles $\int_a^x f(t)\,dt$, but many problems use an endpoint such as $g(x)$. In that case the accumulated total changes for two reasons: the endpoint moves, and the speed of that movement is $g'(x)$. The derivative is therefore $f(g(x))g'(x)$, which is the ordinary chain rule applied to an accumulation function.

Lower limits contribute a minus sign because moving the lower endpoint to the right removes accumulation rather than adding it. With both endpoints variable, differentiate the upper contribution and subtract the lower contribution. A reliable approach is to name the accumulation function first, apply the Fundamental Theorem, and then apply the chain rule visibly rather than attempting to remember a pile of signs from memory.