x^x=10 solve x

不用numerical method 如何 find the approximation ?
答案是x=2.50618414558877。用牛頓法解的
http://www.math.sc.edu/cgi-bin/sumcgi/Newton.pl 非常好用
我好樂意講解其機理﹐至少我們可以識多些東西

圖片參考：http://mathworld.wolfram.com/images/equations/NewtonsMethod/NumberedEquation5.gif

所以重點是找
y=x^x的導數
以下是證明
If y = x^ x and x > 0 then ln y = ln (x^ x)
Use properties of logarithmic functions to expand the right side of the above equation as follows.
ln y = x ln x
We now differentiate both sides with respect to x, using chain rule on the left side and the product rule on the right.
y '(1 / y) = ln x + x(1 / x) = ln x + 1 , where y ' = dy/dx
Multiply both sides by y
y ' = (ln x + 1)y
Substitute y by x^ x to obtain
y ' = (ln x + 1)x^ x
所以
Xn+1=Xn-( Xn ^ Xn -10)/[(ln Xn +1)( Xn ^ Xn)]
代x0=3
3 2.699978108
2.699978108 2.541664405
2.541664405 2.507493579
2.507493579 2.506185967
2.506185967 2.506184146
2.506184146 2.506184146

你要的是不是開方法？