
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 1st, 2019, 09:57 PM  #1 
Newbie Joined: Aug 2018 From: România Posts: 29 Thanks: 2  Derivatives of a function
Hello, Calculate the derivatives of the function: 1) $\displaystyle f(x)=x!$ 2) $\displaystyle f(x)=(2\sqrt{2})\cdot (2\sqrt[3]{2})\cdot \dots \cdot (2\sqrt[x] {2})$. All the best, Integrator 
February 1st, 2019, 10:59 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,373 Thanks: 2010 
Neither function is continuous, so neither function is differentiable.

February 2nd, 2019, 12:54 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,704 Thanks: 670 
$x!=\Gamma(x+1)=\int_0^\infty u^xe^{u}du$. You can get the derivative from this. The second expression is defined only for integers, so you can't get a derivative.

February 2nd, 2019, 03:31 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
On the other hand, the second function, which also can be analytically extended from the positive integers, has a natural choice for this extension which makes its derivative unique. This choice comes from the continuous extension of $f(x) = a^x$ from the integers to the rationals. This automatically determines a unique extension to the positive reals and this extension happens to be analytic. In a nutshell, the first expression has no "correct" answer without more assumptions and the second one does.  
February 2nd, 2019, 09:45 PM  #5  
Newbie Joined: Aug 2018 From: România Posts: 29 Thanks: 2  Quote:
1) From "WolframAlpha" reading: https://www.wolframalpha.com/input/?i=x!%27. Is it correct what WolframAlpha says? 2) I do not know how to use "WolframAlpha" for the first derivative of the function $\displaystyle f(x)=(2\sqrt{2})\cdot (2\sqrt[3]{2})\cdot \dots \cdot (2\sqrt[x]{2})$. Thank you very much! All the best, Integrator  
February 3rd, 2019, 04:46 AM  #6  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,079 Thanks: 845 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  

Tags 
derivatives, function 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
continuity and partial derivatives of a function  robinhg  Real Analysis  1  February 21st, 2014 01:11 PM 
Derivatives: Exponential Function  li1995717  Calculus  2  October 24th, 2013 02:19 AM 
Second derivatives of implicit function  OriaG  Calculus  2  May 25th, 2013 02:56 PM 
Drawing 1st and 2nd derivatives of a given function  AFireInside  Calculus  1  August 26th, 2011 04:18 AM 
smooth function and its derivatives  regfor3  Real Analysis  3  November 10th, 2008 06:53 AM 