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 x₀.

One way to estimate it is to evaluate f at two points, x₁ and x₂, and examine the slope of the line from (x₁, f(x₁)) to (x₂, f(x₂)). But what should we use for x₁ and x₂ and what will we learn about f '(x₀)?

The choice that first occurs to people is to set x₁ = x₀, and x₂ = x₀ + 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.