Editing Difference engine (section)

=== Use of derivatives ===
Many commonly used functions are [[analytic function]]s, which can be expressed as [[power series]], for example as a [[Taylor series]]. The initial values can be calculated to any degree of accuracy; if done correctly the engine will give exact results for first N steps. After that, the engine will only give an [[approximation]] of the function.

The Taylor series expresses the function as a sum obtained from its [[derivative]]s at one point.  For many functions the higher derivatives are trivial to obtain; for instance, the [[sine]] function at 0 has values of 0 or <math>\pm1</math> for all derivatives. Setting 0 as the start of computation we get the simplified [[Maclaurin series]]
:<math>
\sum_{n=0}^{\infin} \frac{f^{(n)}(0)}{n!}\  x^{n}
</math>

The same method of calculating the initial values from the coefficients can be used as for polynomial functions. The polynomial constant coefficients will now have the value
:<math>
a_n \equiv \frac{f^{(n)}(0)}{n!}
</math>