The derivative of a function at a point is the value approached by the average rate of change of the function in an infinitesimally small interval around that point. It represents the slope of the tangent line to the curve of the function at that point and provides information about how the function changes locally.
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$