October 31st, 2017, 10:40 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 205 Thanks: 26  nth derivative
Check if this holds true .... (post your check) Nth derivative example without inspection of first , second , third ... derivative $\displaystyle y=xe^x \; $ $$ (uv)^{(n)}= \sum \limits_{k=0}^{n} {n \choose k} u^{(nk}v^{(k)} \; $$ set $\displaystyle u=x , v=e^x$ $$ y^{(2n)}= \sum \limits_{k=0}^{2n} {2n \choose k} x^{(2nk)}(e^x)^{(k)}=e^x \sum \limits_{k=0}^{2n} {2n \choose k} x^{(2nk)}= {2n \choose 0} x^{(2n)}+{2n \choose 1} x^{(2n1)}+...+{2n \choose 2n1} x^{(1)}+{2n \choose 2n} x^{(0)} $$ $\displaystyle x^{(2nk)}=0 \; , \forall k\in N$ $\displaystyle {(xe^x)}^{(2n)}=e^x (2n+x)=2ne^x +xe^x$ $\displaystyle {(xe^x)}^{n}=ne^x+xe^x$ Last edited by idontknow; October 31st, 2017 at 10:42 AM. 
October 31st, 2017, 11:48 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,791 Thanks: 923 
I'm not really sure what you are asking but you can easily confirm your final statement using induction.

October 31st, 2017, 07:34 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 320 Thanks: 164 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
Take $n =2$ and compute this by hand to see the problem.  
October 31st, 2017, 07:49 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 1,791 Thanks: 923  
November 1st, 2017, 05:06 AM  #5  
Senior Member Joined: Sep 2016 From: USA Posts: 320 Thanks: 164 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
\[(uv)^{(n)} = \sum_{j=0}^n \binom{n}{j}u^{(j)}v^{(nj)}\]  
November 1st, 2017, 11:54 AM  #6  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,976 Thanks: 807  Quote:
When n= 2 that formula is $\displaystyle (uv)''= uv''+ 2u'v'+ u''v$. By the product rule, $\displaystyle (uv)'= u'v+ uv'$ and then $\displaystyle (uv)''= (u'v)'+ (uv')'= (u''v+ u'v')+ (u'v'+ uv'')= u''v+ 2u'v'+ u''v$ so it certainly is true when n= 2. Last edited by Country Boy; November 1st, 2017 at 11:57 AM.  
November 1st, 2017, 04:28 PM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 320 Thanks: 164 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 

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