All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math > Calculus
e^(e^x) (Posted on 2011-02-08) Difficulty: 3 of 5
Find a formula, involving n, the number of e's present in the below function, for the derivative of :

See The Solution Submitted by Math Man    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
possible solution | Comment 3 of 5 |

let f(n,x) be defined recursively as
now we want to find df/dx
d/dx f(n,x) = d/dx e^f(n-1,x)
f'(n-1,x) * e^f(n-1,x)
thus if we define g(n,x) = d/dx f(n,x)
g(n,x) = g(n-1,x) * e^f(n-1,x)
with g(0,x)=1=e^(0)
so now I will prove by induction on n that
g(n,x) = e^( sum f(t,x) for t=0 to n-1 )
now for n=0 this holds
assume it holds for n=k
g(k+1,x) from above is equal to
g(k,x) = e^(sum f(t,x) for t=0 to k-1)
g(k+1,x) = e^(sum f(t,x) for t=0 to k)
thus by the induction hypothesis it holds for all n>=0

so in simplier terms, the derivative is equal to e to the power of the sum of all the smaller towers. 

  Posted by Daniel on 2011-02-08 15:59:41
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information