A derived quantity inherits uncertainty from the measured quantities used to calculate it. Suppose $$y=f(x_1,\ldots,x_n)$$ and the input estimates have small uncertainties. A first-order Taylor expansion gives the sensitivity of $y$ to each input: $$dy\approx\sum_i \frac{\partial f}{\partial x_i},dx_i.$$

If the input uncertainties are independent, the corresponding standard uncertainty is approximately $$u_y^2\approx\sum_i\left(\frac{\partial f}{\partial x_i}\right)^2u_{x_i}^2.$$ For correlated inputs, covariance terms must also be included: $$u_y^2\approx \nabla f^{\mathsf T}\Sigma\nabla f.$$

As an example, the area of a rectangle is $A=LW$. If $L=2.00\pm0.02,\mathrm{m}$ and $W=3.00\pm0.03,\mathrm{m}$ with independent standard uncertainties, then $$u_A^2=W^2u_L^2+L^2u_W^2.$$ Thus $$u_A=\sqrt{(3.00)^2(0.02)^2+(2.00)^2(0.03)^2}\approx0.085,\mathrm{m^2}.$$ The result is approximately $A=6.00\pm0.09,\mathrm{m^2}$.

For products and powers, relative uncertainty often gives a useful interpretation. If $y=x^a z^b$ and the inputs are independent, then $$\left(\frac{u_y}{y}\right)^2\approx a^2\left(\frac{u_x}{x}\right)^2+b^2\left(\frac{u_z}{z}\right)^2.$$

Linear propagation is an approximation. Strong nonlinearities, large uncertainties, bounded variables, or non-Gaussian distributions may require direct propagation by simulation or a more complete probabilistic model. The essential idea is unchanged: uncertainty in a result depends not only on uncertainty in the inputs but also on how sensitively the result responds to them.