In my course we have stated and used the error propagation formula:
$|y-y_0|\approx|f^\prime(x)|\cdot|x-x_0|$
But it was presented with no proof and I wonder if you can help me understand the formula holds?
In my course we have stated and used the error propagation formula:
$|y-y_0|\approx|f^\prime(x)|\cdot|x-x_0|$
But it was presented with no proof and I wonder if you can help me understand the formula holds?
Possibly the best way to understand it is via the mean value theorem: $f(x)-f(x_0)=f'(c)\cdot(x-x_0)$ for some $c$ between $x_0$ and $x$. If $f'$ is continuous, $f'(c)$ can be expected to be close to $f'(x)$ or $f'(x_0)$ when $\lvert x-x_0\rvert$ is small enough.