Thread: Unusual differentiation View Single Post
2004-08-14, 22:11   #3
JuanTutors

Mar 2004

509 Posts

Quote:
 Originally Posted by mfgoode How would you differentiate y = x^( x ) ^(x^)----------x----right up to infinity ? In other words y =X to the power of x to the power of x---up to infinity ? Mally
For 0<x<1, this can't converge to 1. notice that y=x^y. So if, say for x=1/2, we would have that 1=y=(1/2)^1=1/2.

I'm not much into complex analysis, so I don't know all the places that this function converges. However, here is a hint on taking the derivative:

Let f[x] := x^f[x], where x^f[x] is defined as x^f[x] := exp[f[x]*Log[x]], i.e. the number e=2.71828... to the power of f[x]*Log[x], where Log[x] is the natural log. Take the derivative, and solve for df/dx

This has a fairly nice answer, and is related to the Lambert W-function. (I really don't know anything about it, though )