Quote:
Originally Posted by mfgoode
How would you differentiate
y = x^( x ) ^(x^)xright up to infinity ?
In other words y =X to the power of x , to the power of xup to infinity ?
Mally

Thank you one and all for the interest shown in this problem and the keen insight in tackling it.
The Solution: Let Y = x^x^x^X to infinity.
Therefore y =x^y  0 <x<=1
Taking logs logy =ylogx
Hence log y/y=logx
Hence dx/dy = [ y*(1/y) logy*1]/ (y^2) Quotient rule y not=0
Therefore dy/dx= (y^2)/ 1logy
Now y tends to 1 in this range as given above
Therefore dy/dx = 1 as log1 =0
This gives the slope as 1 (tan inverse 1=45*) i.e. m=1
Hence this is the eqn. of a straight line bisecting the angle (90*) in 1st. quadrant
therefore eqn of line is y=x [y=mx+c and c=0]
I remain open for further discussion on this problem
Mally