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I need to find: $$\frac{d}{dx} x!$$

Is there a way to find this? If not, is there a proof that shows we cannot find it?

The graph of $x!$ Function

https://www.desmos.com/calculator/kwfazibe1r

Kenta S
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    You have to deal with Gamma function. – Gonzalo Benavides Jul 02 '18 at 16:57
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    what is 1.02! ? – Vasili Jul 02 '18 at 16:58
  • @GonzaloAlejandroBenavidesGa gamma function? Is that something a 10th standard person can understand? –  Jul 02 '18 at 16:58
  • @Vasya can't that be found out by graphing? –  Jul 02 '18 at 17:00
  • @MathLover Shashwat Asthana has indicated (in the comments above) unfamiliarity with the gamma function, so it's unlikely a question referencing it in the title would be helpful. – BallBoy Jul 02 '18 at 17:08
  • @Vasya https://www.desmos.com/calculator/kwfazibe1r go to this link. –  Jul 02 '18 at 17:11
  • @ShashwatAsthana How does the computer generate that graph? – BallBoy Jul 02 '18 at 17:15
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    @ShashwatAsthana Your link to Desmos shows that Desmos knows about the Gamma function (that fills in the factorial function between the integers). But that function does not have a definition you are likely to understand at this point in your career. But you did ask a good question. – Ethan Bolker Jul 02 '18 at 17:16
  • @ShashwatAsthana: Usually it's the other way around: to graph a function, we take a subset of values from function domain and calculate corresponding values in the function range. These value pairs are points that form the graph. I was just trying to find out your definition of the factorial function. – Vasili Jul 02 '18 at 17:25
  • @YForman the link by MathLover specifies "without using the Gamma function". The question takes lengths to specifically avoid the Gamma function. So, yes, it is a duplicate, as is the one I found. – SlipEternal Jul 02 '18 at 17:26

1 Answers1

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The factorial function is defined only on the domain of the natural numbers (so, e.g., you can't take $\frac12!$). The notion of derivative doesn't make much sense in this domain -- think about the definition of derivative in terms of limits, or think about the intepretation as the slope of the tangent line; how could you have a tangent line if the graph of the factorial function will just be a bunch of isolated points?

In order to make sense of a derivative of the factorial, you'd have to extend the function so that it is defined over a larger domain. The most common way to do this is via the gamma function, which was mentioned in the comments.

BallBoy
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