let f(x) be a function that is defined and differentiable at a point x=a. Then the derivative of f(x) at x=a, denoted by f'(a), is defined as the limit of the ratio of the change in the function value to the change in the input value as the latter change approaches zero:
f'(a) = lim (f(x) - f(a))/(x - a) x→a x→a