Geometric Definition of Derivative
The derivative of a function f (x) at x = x 0 , denoted f'(x 0) or (x 0) , can be naively defined as the slope of the graph of f at x = x 0 . 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 x 0 to be the slope of the tangent line to the graph at x 0 . This tangent line can be thought of in a couple of ways:
- as the line that most closely approximates the graph near the point (x 0, f (x 0)) .
- as the "limit" of the secant lines through (x 0, f (x 0)) and nearby points (x, f (x)) as x approaches x 0 .
In order for the tangent line to be well-defined, the graph of f at x 0 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 x 0 , we say f is differentiable at x 0 . 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 .