Numerical Differentiation

Approximate numerical formulae can be derived using Taylor series.

First Order Derivative

Taylor series expansion of around , truncated after derivative.

Here is a small positive number, and is called the step size.

Error is bounded by :

Forward difference formula

Here is a bound on between and .

Backward difference formula

Here is a bound on between and .

Centered difference formula

Aka. three-point mid point formula. Truncated after derivative.

Error is in order of which is better than .

Proof Hint

• Taylor series expansion of and around are truncated after derivative.
• The below equation is derived by subtracting the two.

Higher Order Derivatives

Taylor series is truncated after derivative.

Second forward difference formula

Second backward difference formula

Second centered difference formula

Accuracy

The accuracy of these divided difference formulas can be increased by including additional terms in the Taylor series. In each step the accuracy is improved by a power of .