What is the derivative of y = x^(x^x)?

We have to differentiate y =
x^(x^x)

y = x^(x^x)

ln y =
ln[x^(x^x)]

=> ln y =
(x^x)*ln(x)

take the derivative of both the sides with
respect to x

=> (1/y)*dy/dx = (x^x)/x + (ln
x)*(x^x)'

let z = x^x, ln z = x*ln
x

1/z(dz/dx) = x/x + ln
x

=> dz/dx = x^x + x^x*ln
x

=> (1/y)*dy/dx = (x^x)/x + (ln x)*(x^x*ln
x+x^x)

=> dy/dx = y[(x^x)/x + (ln x)*(x^x*ln x +
x^x)]

=> dy/dx = x^(x^x)[(x^x)/x + (ln x)*(x^x*ln x
+ x^x)]

The required derivative is dy/dx =
x^(x^x)[(x^x)/x + (ln x)*(x^x*ln x + x^x)]