Fundamental Theorem of Calculus, Part I

Suppose $A(x)=\int_a^x f(t)\,dt$. Increasing $x$ by a small amount $h$ adds a thin strip whose width is $h$ and whose height is close to $f(x)$. The added accumulation is therefore approximately $f(x)h$, so the quotient $[A(x+h)-A(x)]/h$ is approximately $f(x)$. In the limit, the approximation becomes exact under the theorem's hypotheses.

This result is profound because it says accumulation has a local rate, and that rate is precisely the function being accumulated. It is the mathematical statement that a running total changes at the current input rate. The theorem turns area and accumulation functions into differentiable objects and explains why integration and differentiation are inverse processes rather than unrelated units placed next to each other by tradition.