In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or subroutine using values of the function and perhaps other knowledge about the function.

The simplest method is to use finite difference approximations.

A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). Choosing a small number h, h represents a small change in x, and it can be either positive or negative. The slope of this line is $${\displaystyle {\frac {f(x+h)-f(x)}{h}}.}$$ This expression is Newton's difference quotient (also known as a first-order divided difference).

To obtain an error estimate for this approximation, one can use Taylor expansion of ${\displaystyle f(x)}$ about the base point ${\displaystyle x}$ to give $${\displaystyle f(x+h)=f(x)+hf'(x)+{\frac {h^{2}}{2}}f''(c)}$$ for some ${\displaystyle c}$ between ${\displaystyle x}$ and ${\displaystyle x+h}$. Rearranging gives $${\displaystyle f'(x)=\underbrace {\frac {f(x+h)-f(x)}{h}} _{\text{Slope of secant line}}-\underbrace {{\frac {h}{2}}f''(c)} _{\text{Error term}}.}$$ The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line and the error term vanishes. Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: $${\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}$$

Since immediately substituting 0 for h results in ${\displaystyle {\frac {0}{0}}}$ indeterminate form, calculating the derivative directly can be unintuitive.

Equivalently, the slope could be estimated by employing positions x − h and x.

Another two-point formula is to compute the slope of a nearby secant line through the points (x − h, f(x − h)) and (x + h, f(x + h)). The slope of this line is $${\displaystyle {\frac {f(x+h)-f(x-h)}{2h}}.}$$

This formula is known as the symmetric difference quotient. In this case the first-order errors cancel, so the slope of these secant lines differ from the slope of the tangent line by an amount that is approximately proportional to ${\displaystyle h^{2}}$. Hence for small values of h this is a more accurate approximation to the tangent line than the one-sided estimation. However, although the slope is being computed at x, the value of the function at x is not involved.