Finite Differences

Numerical Analysis · Axiom Academy

The foundation of numerical differentiation The derivative at a point x is defined as the limit of a difference quotient: This limit represents the slope of the tangent line. The key insight: if we don't take the limit (use a small but finite h), we get an approximation! The forward difference approximates the derivative using the point ahead: This looks at the secant line from x to x+h. It's called "forward" because we step forward by h. The backward difference uses the point behind: This looks at the secant line from x-h to x. Useful when data ahead isn't available (like at the end of a time series). The central difference uses points on both sides: By averaging the forward and backward slopes, errors tend to cancel out. This is typically more accurate than one-sided differences! Let's compare all three methods at once:

This is the written version of the interactive lesson above. See the full Numerical Analysis course.