Let $g$ be differentiable on $[a,b]$ and $g'$ continuous on this interval. Let $f$ be continuous on the range of $g$.
Call $A(x) = \int_{a}^{x} f(g(t))g'(t)dt$ and $B(x) = \int_{g(a)}^{g(x)} f(t)dt$
Why is it that $A$ and $B$ are differentiable and how can we prove that $A'(x) = B'(x), \forall x \in [a,b]$? My ideas are using the definition of differentiability but I'm a bit intimidated to use integrals inside limits so I'm not entirely sure where to start. I believe this serves as a lemma to the substitution rule.
By the fundamental theorem of Calculus,$$A'(x)=f\bigl(g(x)\bigr)g'(x).$$On the other hand, if $C(x)=\int_{g(a)}^xf(t)\,\mathrm dt$, then, for the same reason, $C'(x)=f(x)$, and, since $B=C\circ g$, the chain rule tells you that$$B'(x)=C'\bigl(g(x)\bigr)g'(x)=f\bigl(g(x)\bigr)g'(x).$$