How do you differentiate this simple problem:

f(x) = 1/ (x^x)
first derivative please, with working!

... f(x) = 1/ x^(x)
or y = x^(-x)
or ln(y) = -x ln(x)
or 1/y dy/dx = - ln(x) - 1
or dy/dx = - y ( ln(x) + 1 )
or dy/dx = - ( ln(x) + 1 ) / x^(x)