= Noise amplification by differentiation of a linear interpolant
{title2=$\Delta p(x)=\delta x/h,\qquad\Delta p'(x)=\delta/h$}
For <linear interpolation> at zero and $h>0$, an error $\delta$ in the second sampled value, with the first exact, changes the <linear interpolant> by $\Delta p(x)=\delta x/h$. On $[0,h]$, its <uniform norm> is exactly $|\delta|$, whereas the <derivative> change is the constant $\delta/h$. With independent absolute bounds $|\delta_0|,|\delta_1|\leq\delta$ on both samples, the value change remains at most $\delta$ and the <derivative> change is at most $2\delta/h$, attained for opposite endpoint errors.
Thus bounded absolute sample noise can remain bounded under value <interpolation> while becoming unbounded under <numerical differentiation> as $h\to0$. Fixed nonzero sample noise also prevents exact convergence of the value approximation; stability against perturbations and convergence to exact data are different assertions. For a second-sample error $a\epsilon(h)$ with fixed $a>0$, value convergence requires $\epsilon(h)\to0$, while <derivative> convergence requires $\epsilon(h)=o(h)$ in <little o notation>, in addition to vanishing <truncation error>.
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