The Fundamental Theorem of Calculus is one of the most amazing and important theorems in analysis. It is a non-trivial result that links the concept of area and gradient, two seemingly unrelated concepts.

## Fundamental Theorem of Calculus

The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

## First part

Let be a continuous real-valued function defined on a closed interval . Let be the function defined, for all in , by

Then is uniformly continuous on , differentiable on the open interval , and for all in .

## Second part

Let and be real-valued functions defined on such that is continuous and for all ,

If is Riemann integrable on , then