April 23rd, 2018, 10:20 PM  #1 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2  Calculus  Chain Rule
Let f1(x) = e^x, f2(x) = e^f1(x) and generally fn+1(x) = e^fn(x) for all n . For any fixed n, Find the the derivative of fn(x) Please help Last edited by arybhatta01; April 23rd, 2018 at 10:22 PM. 
April 23rd, 2018, 11:21 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,825 Thanks: 1051 Math Focus: Elementary mathematics and beyond 
Hi arybhatta01. Have you made any progress solving this on your own? If so, can you post your work? If not, any thoughts on where to begin?

April 24th, 2018, 10:14 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,248 Thanks: 887 
You mention the chair rule. If $\displaystyle f_{n+1}(x)= e^{f_n(x)}$, then, using the chain rule, $\displaystyle f_{n+1}'= e^{f_n(x)}(f_n(x)')$. I would be inclined to find the derivatives of the first three or four function, make a "guess" at derivative of the nth function, then try to prove that "guess" using induction.
Last edited by skipjack; April 24th, 2018 at 07:06 PM. 
April 24th, 2018, 10:53 PM  #4 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2 
Yes . It worked . We need to use chain rule to get the derivative of the 1st 3 function f1 f2,f3 and the use induction to get fn . Thanks.


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