Question

What is the differentiation of f(x) = (e^x)^(e^x)

Answer 1

let u = e^x
then y = u^u
ln both sides lny = ln u^u = u ln u
derivative both sides \[\frac{y'}{y} = u'ln u+(ln u)'u\]

Answer 2

\[f \left( x \right)=\left(e ^{x} \right)^{\left( e ^{x} \right)}\]
\[\ln f \left( x \right)=\ln \left( e ^{x} \right)^{\left( e ^{x} \right)}=e ^{x}\ln e ^{x}=xe ^{x}\]
\[\frac{ f \prime \left( x \right) }{ f \left( x \right) }=xe ^{x}+1*e ^{x}=\left( x+1 \right)e ^{x}\]
\[f \prime \left( x \right)=\left( x+1 \right)e ^{x}f \left( x \right)=\left( x+1 \right)e ^{x}\left( e ^{x } \right)^{\left( e ^{x} \right)} \]