Editing Condition number (section)

=== One variable ===

The ''absolute'' condition number of a [[differentiable function]] <math>f</math> in one variable is the [[absolute value]] of the [[derivative]] of the function:

: <math>\left|f'(x)\right|</math>

The ''relative'' condition number of <math>f</math> as a function is <math>\left|xf'/f\right|</math>. Evaluated at a point <math>x</math>, this is

: <math>\left|\frac{xf'(x)}{f(x)}\right|=\left|\frac{(\log f)'}{(\log x)'}\right|.</math>

Note that this is the absolute value of the [[Elasticity (economics)|elasticity]] of a function in economics.

Most elegantly, this can be understood as (the absolute value of) the ratio of the [[logarithmic derivative]] of <math>f</math>, which is <math>(\log f)' = f'/f</math>, and the logarithmic derivative of <math>x</math>, which is <math>(\log x)' = x'/x = 1/x</math>, yielding a ratio of <math>xf'/f</math>. This is because the logarithmic derivative is the [[Infinitesimal calculus|infinitesimal]] rate of relative change in a function: it is the derivative <math>f'</math> scaled by the value of <math>f</math>. Note that if a function has a [[zero of a function|zero]] at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to positive or negative, yielding a ratio with zero in the denominator, hence infinite relative change.

More directly, given a small change <math>\Delta x</math> in <math>x</math>, the relative change in <math>x</math> is <math>[(x + \Delta x) - x] / x = (\Delta x) / x</math>, while the relative change in <math>f(x)</math> is <math>[f(x + \Delta x) - f(x)] / f(x)</math>. Taking the ratio yields

: <math>\frac{[f(x + \Delta x) - f(x)] / f(x)}{(\Delta x) / x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{(x + \Delta x) - x} = \frac{x}{f(x)} \frac{f(x + \Delta x) - f(x)}{\Delta x}.</math>

The last term is the [[difference quotient]] (the slope of the [[secant line]]), and taking the [[limit (mathematics)|limit]] yields the derivative.

Condition numbers of common [[elementary function]]s are particularly important in computing [[significant figures]] and can be computed immediately from the derivative. A few important ones are given below:
{| class="wikitable"
! Name || Symbol || Relative condition number
|-
| Addition / subtraction || <math>x + a</math> || <math>\left|\frac{x}{x+a}\right|</math>
|-
| Scalar multiplication || <math>a x</math> || <math>1</math>
|-
| Division || <math>1 / x</math> || <math>1</math>
|-
| [[Polynomial]] || <math>x^n</math> || <math>|n|</math>
|-
| [[Exponential function]] || <math>e^x</math> || <math>|x|</math>
|-
| [[Natural logarithm]] function || <math>\ln(x)</math> || <math>\left|\frac{1}{\ln(x)}\right|</math>
|-
| Sine function || <math>\sin(x)</math> || <math>|x\cot(x)|</math>
|-
| Cosine function || <math>\cos(x)</math> || <math>|x\tan(x)|</math>
|-
| Tangent function || <math>\tan(x)</math> || <math>|x(\tan(x)+\cot(x))|</math>
|-
| Inverse sine function || <math>\arcsin(x)</math> || <math>\frac{x}{\sqrt{1-x^2}\arcsin(x)}</math>
|-
| Inverse cosine function || <math>\arccos(x)</math> || <math>\frac{|x|}{\sqrt{1-x^2}\arccos(x)}</math>
|-
| Inverse tangent function || <math>\arctan(x)</math> || <math>\frac{x}{(1+x^2)\arctan(x)}</math>
|}