In the previous section, we learned that the Fundamental Theorem of Calculus says that if \(f\) is continuous on \([a,x]\), then \[\bbox[#F2F2F2,5px,border:2px solid black]{\frac{d}{d{\color{red}x}}\int_{a}^{{\color{red}x}}f({\color{blue}t})d{\color{blue}t}=f({\color{red}x})}\]
We may use the chain rule in conjuction with the first part of the Fundamental Theorem of Calculus to find the derivative if the upper limit or the lower limit integration is a function. For example, if \[F(x)=\int_{a}^{g(x)}f(t)dt,\] to find \(F'(x)\), let \(u=g(x)\). Then
\[\begin{aligned}
\frac{dF}{dx} & =\frac{dF}{du}\frac{du}{dx}\\
& =\left(\frac{d}{du}\int_{a}^{u}f(t)dt\right)g'(x)\\
& =f(u)g'(x) &&{\small \text{ (by the Fundamental Theorem of Calculus)}}\\
& =f(g(x))g'(x) &&{\small (u=g(x))}.
\end{aligned}\]