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

## 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

$f ( x 0 + d ) − f ( x 0 ) d$

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.