Estimating an error

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Suppose that you measure three quantities $x$ , $y$ and $z$ and need the composite variable $w=ax+ by+ cz$ with $a$ , $b$ and $c$ . Suppose your measurements are $x_m\pm {\vartriangle}x$ , $y_m\pm {\vartriangle}y$ and $z_m\pm {\vartriangle}z$ . What is the estimated error of $w$ ?
Denote the errors as differentials, so $\dd x$ rather than ${\vartriangle}x$ , and so on.

$\dd w={}$

Functions of several variables: Total differential and Taylor approximation

Estimating an error