Why is the derivative of e^(x^x) multiplied by x^x?

[royisher]

Homework Statement

Problem: e(x^x)

The Attempt at a Solution

I came up with - f'(x) = e(x^x)[ln(x)+1]. The solution should be - x^x*e(x^x)[ln(x)+1] . Can someone please help me understand why should I add x^x to the solution.

Much appreciated!

 

[dextercioby]

You should use chain rule and logarithmic differentiation. First use the chain rule.

 

[royisher]

Aha, so I should just apply it like that -

f(x) = e(x^x) -> f'(x) = e(x^x)

g(x) = x^x -> ln(g(x))=x*ln(x) -> g'(x)=x^x(ln(x)+1)

(f°g)'(x) = e(x^x)*x^x(ln(x)+1)

Thank you very much!