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Concept
Numerical differentiation
Summary
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function 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
This expression is Newton's difference quotient (also known as a first-order divided difference).
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. 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:
Since immediately substituting 0 for h results in 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
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 . 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
where is some point between and .
This error does not include the rounding error due to numbers being represented and calculations being performed in limited precision.
Official source
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