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

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

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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

\frac{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.