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In a 2003 paper in French, "Estimation par maximum de vraisemblance de la dérivée d’un signal bruité. Application à la caractérisation de vérins pneumatiques" (Maximum likelihood estimation of the derivative of a noisy signal. Application to the characterization of pneumatic cylinders) [from the early GRETSI french-speaking conference on signal and image ...


Your original differentiator, which should be $x(n)-x(n-1)$, is called a "first difference" differentiator. That differentiator amplifies high-frequency noise. As a next step I suggest you try what's called the "central difference" differentiator defined by: $Diff = \frac{x(n)-x(n-2)}{2} $ which does not amplify high-frequency noise.

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