Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site alice.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!allegra!alice!apdoo
From: apdoo@alice.UUCP (Alan Weiss)
Newsgroups: net.puzzle
Subject: Re: More on Interview Questions.
Message-ID: <3488@alice.UUCP>
Date: Mon, 25-Mar-85 10:02:54 EST
Article-I.D.: alice.3488
Posted: Mon Mar 25 10:02:54 1985
Date-Received: Tue, 26-Mar-85 05:35:15 EST
References: <379@cavell.UUCP>, <917@utcsri.UUCP>
Organization: Bell Labs, Murray Hill
Lines: 23

This line is for the line eater.

	The solution given to the question "What is the derivative of x! ?"
is not quite right.  It is true that

x! = integral (t^x exp(-t) dt)

but differentiating with respect to x we get

derivative of x! = integral (log(t)*t^x*exp(-t))dt

which is not, in general, equal to x!.  In fact, it is not hard to show
that the only solution of the equation

df/dx = f

is f = C exp(x), where C is an arbitrary constant.  Parenthetically,
it is not hard to show that

derivative of x! is approximately equal to   x!*log(x).

See, for example, references to the diagamma function or any book on
asymptotic expansions of integrals.