In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or subroutine using values of the function. Unlike analytical differentiation, which provides exact expressions for derivatives, numerical differentiation relies on the function's values at a set of discrete points to estimate the derivative's value at those points or at intermediate points. This approach is particularly useful when dealing with data obtained from experiments, simulations, or situations where the function is defined only at specific intervals.
Finite differences
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
f ( x + h ) − f ( x ) h . {\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 f ( x ) {\displaystyle f(x)} about the base point x {\displaystyle x} to give
f ( x + h ) = f ( x ) + h f ′ ( x ) + h 2 2 f ″ ( c ) {\displaystyle f(x+h)=f(x)+hf'(x)+{\frac {h^{2}}{2}}f''(c)}
for some c {\displaystyle c} between x {\displaystyle x} and x + h {\displaystyle x+h} . Rearranging gives
f ′ ( x ) = f ( x + h ) − f ( x ) h ⏟ Slope of secant line − h 2 f ″ ( c ) ⏟ Error term . {\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:
f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h . {\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}
Since immediately substituting 0 for h results in 0 0 {\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
f ( x + h ) − f ( x − h ) 2 h . {\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 h 2 {\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. The estimation error is given by
R = − f ( 3 ) ( c ) 6 h 2 , {\displaystyle R={\frac {-f^{(3)}(c)}{6}}h^{2},}
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