The deriviative of f(x) at x=a is given by
The quotient represents the slope of the line (called the secant line) through the points (a, f(a)) and (a+h, f(a+h)):
Consider the function f(x) = x2 at x = 0.5:
Let's calculate the slope of the secant line for h = -1.5, -1.0, and -0.5:
These secant lines (and their slopes) are shown on the following graph:
As h gets smaller and smaller (approaching 0 from the left), the secant line pivots around and gets closer and closer to the tangent line shown in red. That is,
Let's calculate the slope of the secant line for h = 1.5, 1.0, and 0.5:
These secant lines (and their slopes) are shown on the following graph:
As h gets smaller and smaller (approaching 0 from the right), the secant line gets closer and closer to the tangent line shown in red. That is,
Since both one-sided limits exist and are equal we can assert that
Therefore, the derivative of f(x) = x2 at x = 0.5 is 1. That is, f '(0.5) = 1. The animated graphic below illustrates the secant lines as h approaches zero from the left and from the right. In both cases, the secant line approaches the tangent line which has a slope of one.
Rather than finding the limit of the slope at some particular value of x, let's find the limit in terms of x:
The slope of the line tangent to the curve f(x) = x2 at any value of x is given by f''(x) = 2x. In the graph below, several lines tangent to the curve f(x) = x2 are shown (with the corresponding slopes). Notice that, in every case, the slope = 2x where x is the x-coordinate of the corresponding point on the curve.