Since the derivative of a function f at any value x0 where the function is differentiable is just a number, f'(x0), we may define the derivative function, denoted f'(x) or (x), to be the function that maps a value x where f is differentiable to the derivative of f at x. Since f' is a function in its own right, we can graph it, just like any other function.

For example, let us calculate the derivative function (also called simply the "derivative") of f (x) = 5x. Since the graph of this function is a line with slope 5, we might guess that the derivative will equal 5 at every point. We verify this with the formula for the derivative of f at a point x:


Indeed, f'(x) = 5, a constant function.