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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See here & here. – J.G. Apr 14 '21 at 21:02
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2What is the definition of derivative? Have you tried to apply it to a factorial? – Rolazaro Azeveires Apr 14 '21 at 21:06
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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).
Karl
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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 $m$ .
MPW
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Hint: $ x! = \Gamma(x+1)$ where $Gamma$ is Euler's gamma function
phaedo
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This does not reply the question, or provide any meaningful hint, does it? – Rolazaro Azeveires Apr 14 '21 at 21:07
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