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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

In general, no. Even interpolating the data doesn't give you any guarantees that you're capturing the behavior of the function.

If I gave you four nails, could you identify the house that it builds?

Fundamentally, this is because a derivative is defined as a limit... without something to tell you how the limit behaves, you cannot estimate the derivative.

That said, there are things you can do if you want to approximate a derivative if you know the properties of the data. For example, most GPS systems record your driving speed to within 1/2 of a mile per hour (or kilometer per hour). Obviously, your speed is continuous, so if you have a sufficiently large set of GPS speed data, you could approximate the acceleration your car undertakes at any point by attempting to smooth your quantized data.

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Good answer, thanks. –  mirai Oct 23 '13 at 20:40

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