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For example, $f(x) = 3x^2$, and I want to obtain $f '(2)$ using only information I have with the point $x = 1$. Is there an equation I could use to do this which would apply for all functions, not just the one I mentioned above?

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    Why would you want to do this? You have $f$. What prevents you from simply taking the derivative?2012-10-11
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    I know I can do this, I was just wondering if there's a relationship between consecutive points involving higher derivatives (2nd derivative and so on).2012-10-11
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    What do you mean by "consecutive points"? For the question you asked in the post, the answer is no. You cannot determine $f'(2)$ knowing only $f(1)$. You can't determine $f'(2)$ knowing every derivative of $f$ at 1. You cannot even determine $f'(2)$ knowing the value of $f(x)$ for all $x \in (-\infty, 3/2)$. If you assume $f$ is analytic or polynomial, things become more interesting.2012-10-13

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