= Sensitivity coefficient
{title2=$c_i=\partial f/\partial x_i$}
A <sensitivity coefficient> describes how an estimated output of a <measurement model> changes when one input changes slightly. For $y=f(x_1,\ldots,x_n)$, the first-order coefficients, evaluated at the input estimates, are
$$
c_i=\frac{\partial f}{\partial x_i},\qquad \delta y\simeq\sum_i c_i\,\delta x_i.
$$
A <sensitivity coefficient> can have a sign and a <unit of measurement>: for an output in ohms and an input in volts, its unit is ohms per volt. The GUM also describes estimating <sensitivity coefficients> experimentally by varying inputs.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections5.1.3–5.1.4]{ref}
In a <measurement uncertainty budget>, <propagation of uncertainty> uses these coefficients to transform input <variances> and <covariances>:
$$
u_y^2\simeq\sum_i c_i^2u_i^2+2\sum_{i<j}c_ic_j\operatorname{Cov}(X_i,X_j).
$$
Here $u_i$ denotes an input <standard uncertainty>. The derivative's sign disappears from its independent <variance> term but still matters in a <covariance> term.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections5.1–5.2, equations10 and13]{ref}
For an original <electrical resistance> example, $R=V/I$, $V=10\,\mathrm V$ and $I=2\,\mathrm A$ give $R=5\,\Omega$ and
$$
c_V=\frac1I=0.5\,\Omega/\mathrm V,\qquad c_I=-\frac{V}{I^2}=-2.5\,\Omega/\mathrm A.
$$
Independent input <standard uncertainties> of0.1volt and0.02ampere each contribute0.05ohm on the output scale, giving $u_R=\sqrt{0.05^2+0.05^2}\,\Omega\approx0.0707\,\Omega$. A larger <sensitivity coefficient> alone does not identify the dominant uncertainty: the input <standard uncertainty> and any <correlations> also matter.
The derivative is a local approximation. If the input spread spans a strongly nonlinear part of the <measurement model>, first-order <propagation of uncertainty> can be inadequate even when every derivative has been computed correctly.
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