Fundamental theorem of calculus II

Fundamental theorem of calculus II: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. Suppose $F:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$, differentiable on $(a,b)$, and satisfies $F'(x)=f(x)$ for all $x\in(a,b)$. Then

Combined with fundamental theorem of calculus I, this explains why differentiation rules can be used to evaluate definite integrals via antiderivatives.