From the article on derivative in Wikipedia:
The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)). The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations.
Does this mean that derivatives of functions are just approximations ? (In the sense that the value of $f(x)$ at $a$ is exact -- $f(a)$ -- whereas the derivative $f'(a)$ is not so ?)
EDIT: Based on some of the comments, is it the case then, that the slope of the line passing through $(a,f(a)$ is the derivative and the approximations defined by the secant line slopes approach this real derivative in the limit ?