If the derivative of x factorial exists, what is it? I have tried calculating it on a Derivative Calculator but it doesn't seem to return a result.
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$\begingroup$The factorial function is only defined on nonnegative integers, so it doesn't have a derivative, but its generalization is the gamma function, which has a derivative (see the Wikipedia page).
$\endgroup$ $\begingroup$On the Wikipedia page cited in other answers/comments, one finds the formula
$$\Gamma'(m+1) = m!\left( -\gamma + \sum_{k=1}^{m}\frac{1}{k}\right)$$
for positive integers $m$, where $\gamma$ is the Euler-Mascheroni constant ($\gamma \approx 0.57721$).
Since $m! = \Gamma(m+1)$, one could reasonably call this the derivative of $m!$ with respect to $m$ .
$\endgroup$ $\begingroup$Hint: $ x! = \Gamma(x+1)$ where $Gamma$ is Euler's gamma function
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