The derivative of a function f (x) at x = x0, denoted f'(x0) or (x0), can be naively defined as the slope of the graph of f at x = x0. The problem is that we have not said what we mean by the slope of an arbitrary graph at a point. We do, however, know what we mean by the slope of a line. Therefore, we define the slope of the graph of f at a point x0 to be the slope of the tangent line to the graph at x0. This tangent line can be thought of in a couple of ways:

  1. as the line that most closely approximates the graph near the point (x0, f (x0)).
  2. as the "limit" of the secant lines through (x0, f (x0)) and nearby points (x, f (x)) as x approaches x0.
This second definition is the one we will make rigorous later on as the limit definition of the derivative. The concepts of tangent lines and secant lines are illustrated below.
Figure %: Tangent and Secant Lines

In order for the tangent line to be well-defined, the graph of f at x0 must be sufficiently smooth. Furthermore, the tangent line must not be vertical, for a vertical line is not a function, and cannot be assigned a slope. If the slope of the tangent line, and hence the derivative of f, are well-defined at a point x0, we say f is differentiable at x0. As would be expected, a function that is differentiable at a point must also be continuous at that point. On the other hand, not all functions that are continuous at a point are also differentiable at that point. For example, consider the absolute value function at x = 0.

Figure %: Plot of f (x) = | x|