# Finding derivative at a point in a set

If I have a few values for f(x), i.e. {(0,1), (2, 3), (5, 6)}, is there a way to calculate the derivative at, say f(6), without interpolation?

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No. You cannot. Given any finite set of coordinates, there is a continuous, nowhere-differentiable function with those points on its graph. Moreover, we can construct everywhere infinitely-differentiable functions with those points on its graph to give any derivative value we like at a given finite set of values. (Both of these claims are readily shown using bump functions, and the former claim also uses the fact that there exist continuous nowhere-differentiable functions.)

We actually need to know the function, or at least know how it is defined on a set having the point we're interested in as a limit point.

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Right, that was my intuition but I hoped there was something clever out there. Thanks. –  mirai Oct 23 '13 at 20:39