How is a derivative related to area?

The derivative of a function at a point indicates the rate of change of the function at that point. Its connection to area is expressed through the integral: if you imagine the graph of the function y = f(x), then the area under the graph over the interval [a, b] is equal to the integral ( \int_a^b f(x) dx ).

The Newton-Leibniz theorem links the derivative and the area: if F(x) is an antiderivative of the function f(x), then

[ F'(x) = f(x) ]

and

[ \int_a^b f(x) dx = F(b) - F(a) ]

In other words, the derivative of the function is the instantaneous rate of change of the area under the graph of the function f(x). If you consider the area under the graph as a function of the upper limit of integration, then its derivative at a point is equal to the value of the function f(x) at that point.