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Has anyone here heard of the Derivative existence theorem? Derivative existence theorem: For $f$ defined on some interval including $a$, $f$ is differentiable at $a$ if and only if there exists a function $\varphi $, which is continuous at $a$, and satisfies $\forall x\leq $ interval, $f (x) - f (a) = \varphi (x) (x-a)$.
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