Denote by [itex]V(x)[/itex] the speed of a particle at position x. Let's call [itex]v(x;\zeta)[/itex] a measurement of it, which depends on some parameter [itex]\zeta[/itex], and denote the error by [tex]\epsilon(x;\zeta)=v(x;\zeta)-V(x).[/tex] In order for the measurement to produce meaningful results, we must have some kind of error estimate such that, for any x [tex]|\epsilon(x;\zeta)|\le E(x;\zeta)[/tex] where E is a known positive function, which ideally tends to zero as [itex]\zeta[/itex] tends to zero (we are not considering quantum mechanical effects). My question is: can we obtain a similar estimate for the error in the derivative of [itex]v(x;\zeta)[/itex] (e.g., as a function of [itex]E(x;\zeta)[/itex], [itex]V(x)[/itex] or [itex]V'(x)[/itex]) from the information given above, or do we not have enough information? You can assume that the derivative of [itex]v(x;\zeta)[/itex] is calculated by finite difference, and that the discretization error involved is negligible.