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The antiderivative of a function f, denoted as F, is a function whose derivative is equal to the original function f. Mathematically, if F'(x) = f(x), then F is the antiderivative of f.

The area function A of a function f, calculates the area under the curve of the function f from a fixed starting point a to a variable point x. Mathematically, the area function A(x) is defined as: A(x) = \(\int_a^x f(t) dt\)

Since F'(x) = f(x), we can write the area function as: A(x) = \(\int_a^x F'(t) dt\) According to the Fundamental Theorem of Calculus, if a function has a continuous derivative on a given interval, then the integral of its derivative from a to x is equal to the difference of the function evaluated at those two points. In this case, F(t) has a continuous derivative, which is the original function f(t). Therefore: A(x) = F(x) - F(a) This relationship states that the difference in the antiderivative F of the function f, evaluated at x and at the starting point a, gives the area under the curve of the function f from a to x. Therefore, the functions F and A are related through their derivatives and the Fundamental Theorem of Calculus.