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*|