Finite Difference Calculator

Finite differences approximate derivatives using nearby function values.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

Finite differences replace the limiting definition of a derivative with slopes computed over a small nonzero step h. A forward difference uses f(x) and f(x+h); a backward difference uses f(x−h) and f(x); a central difference uses values on both sides of x.

For a smooth function, the forward and backward formulas have truncation error proportional to h, while the standard central formula has error proportional to h². That is why central differences are often more accurate at the same moderate step size when both neighboring values are available.

Smaller h is not always better on a computer. Very tiny differences subtract nearly equal floating-point numbers, amplifying roundoff. Numerical differentiation therefore balances truncation error against floating-point error.

Match the supplied values to the selected stencil. For a central difference, f(x−h) and f(x+h) must be evaluated at points equally spaced around the same x. For forward or backward differences, the two values must likewise be exactly one step h apart. Using inconsistent spacing invalidates the formula even if the arithmetic is flawless.

Finite differences estimate local slope, not average change over a large interval. If h is too large, curvature can make the secant slope differ noticeably from the derivative at x. Reducing h should initially improve the estimate for a smooth function; if the answer becomes unstable at extremely tiny h, floating-point cancellation is likely dominating.