Introduction

Abstract

The simple approximate formula for the computation of the first derivative of a function y(x),

$$ y'\left( x \right) \approx \frac{1}{{2h}}\left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$

((1.1))

is known to work well when y(x) is smooth enough. However, if y(x) is an oscillatory function of the form

$$ y'\left( x \right) = {f_1}\left( x \right)\sin \left( {\omega x} \right) + {f_2}\left( x \right)\cos \left( {\omega x} \right) $$

((1.2))

with smooth f ₁(x) and f ₂(x), the slightly modified formula

$$ y'\left( x \right) \approx \frac{1}{{2h}} \cdot \frac{\theta }{{\sin \theta }} \cdot \left[ {y\left( {x + h} \right) - y\left( {x - h} \right)} \right], $$

((1.3))

whereθ=ωh, becomes appropriate.