Numerical differentiation

Definition

Numerical differentiation refers to a method for computing the approximate numerical value of the derivative of a function at a point in the domain as a difference quotient. Explicitly, the numerical derivative $f^{\prime }(x)$ of a function $f$ at a point $x$ may be computed using either of these three formulas, for $h$ a sufficiently small positive real number:

Expression	Interpretation of limit as $h\to 0$
Forward difference quotient $\frac{f(x+h)-f(x)}{h}$, comes from the forward difference form of the finite difference	The right-hand derivative $f{\text{'}}_{+}(x)$. If $f$ is differentiable at $x$, this equals the two-sided derivative $f^{\prime }(x)$.
Backward difference quotient $\frac{f(x)-f(x-h)}{h}$, comes from the backward difference form of the finite difference	The left-hand derivative $f{\text{'}}_{-}(x)$. If $f$ is differentiable at $x$, this equals the two-sided derivative $f^{\prime }(x)$.
Central difference quotient $\frac{f(x+h)-f(x-h)}{2h}$, comes from the central difference form of the finite difference	If $f$ is differentiable at $x$, this equals the two-sided derivative $f^{\prime }(x)$. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of $f$.