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}.

Figure %: Tangent and Secant Lines

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.

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

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