







Differentiate y = x^x^x 
20170319 

From Nafis: differentiate y = x^x^x Answered by Penny Nom. 





Differentiate x^x  2^sinx 
20130809 

From tarun: derivative of x^x  2^sinx Answered by Penny Nom. 





The derivative of y=x^x 
20100409 

From David: So, its David, and I was wondering about the derivative of y=x^x. I have often seen it be shown as x^x(ln(x)+1), but when I did it through limits it turned out differently. Here's what I did:
It is commonly know that df(x)/dx of a function is also the limit as h>0 of f(x+h)f(x)/h.
To do this for x^x you have to start with lim h>0 ((x+h)^(x+h)x^x)/h. The binomial theorem then shows us that this is equal to lim h>0 (x^(x+h)+(x+h)x^(x+h1)h+...x^x)/h
This is also equal to lim a>0 lim h>0 (x^(x+a)+(x+h)x^(x+h1)h...x^x)/h.
Evaluating for a=0 you get lim h>0 (x^x+(x+h)x^(x+h1)h...x^x)/h
Seeing as the last 2 terms on the numerator cancel out you can simplify to a numerator with h's is each of the terms, which you can then divide by h to get:
lim h>0 (x+h)x^(x+h1)... which when evaluated for h=0 gives us: x(x^(x1)). This statement is also equal to x^x.
This contradicts the definition of the derivative of x^x that is commonly shown. So, my question is: can you find any flaws in the logic of that procedure? I do not want to be shown how to differentiate x^x implicitly because I already know how to do that. Answered by Robert Dawson. 





x^x = 1 
20090828 

From Waleed: x^x=1 Answered by Robert Dawson. 





The integral of x^x 
20090618 

From ANGIKAR: what would be the integration of (X^Xdx)?
give answer in details. Answered by Robert Dawson and Harley Weston. 





Differentiate y= (x^x^x)^x 
20080627 

From emril: Differentiate y= (x^x^x)^x Answered by Harley Weston. 





differentiate Y=X^X^X 
20040913 

From Kunle: differentiate Y=X^X^X Answered by Penny Nom. 





The derivative of x to the x 
20040214 

From Cher: what about the derivative of x to the power x? Answered by Penny Nom. 





Integrating x^x 
20020618 

From Jeremy: I am a student at the University of Kansas and I am wondering if there is a general antiderivative for x^{ x} (i.e. the integral of x^{ x} dx)? I've looked in a bunch of Table of Integrals and have found nothing (can you guys find it?), so I'm sort of wondering if this may be a research type question. Answered by Claude Tardif. 

