## 7.1 Introduction: the Obvious Approximation: f '(x) ~ (f(x+d) - f(x)) / d

Suppose we have a given function, f, and we seek its **derivative at argument
x**_{0}.

One way to estimate it is to evaluate f at two points, x_{1} and x_{2}, and examine the slope of the line from (x_{1}, f(x_{1}))
to (x_{2}, f(x_{2})). But what should we use for x_{1}
and x_{2} and what will we learn about f '(x_{0})?

The choice that first occurs to people is to set x_{1} = x_{0}, and
x_{2} = x_{0} + d for some very small d. So one can compute

This is not a horrible thing to do, but it is not very good, as we shall see.

**What's wrong with it?**

Well, if d is too big, the linear approximation won't be accurate and if d
is too small, the inaccuracy of your calculation tools may screw up your answer.
And the transition from being too big to too small may be difficult to find.