a x = e x p ( x ⋅ l o g ( a ) ) x x = e x p ( x ⋅ l o g ( x ) ) x ( x x ) = e x p ( x x l o g ( x ) ) d d x x x = e x p ( x ⋅ l o g ( x ) ) ⋅ ( 1 ⋅ l o g ( x ) + 1 ) d d x x ( x x ) = e x p ( x x l o g ( x ) ) ⋅ ( ( d d x x x ) l o g ( x ) + x x − 1 ) d d x x ( x x ) = e x p ( x x l o g ( x ) ) ⋅ ( ( e x p ( x ⋅ l o g ( x ) ) ⋅ ( 1 ⋅ l o g ( x ) + 1 ) ) l o g ( x ) + x x − 1 ) ( x x ) x = e x p ( x ⋅ l o g ( x x ) ) = e x p ( x ( x ⋅ l o g ( x ) ) ) = e x p ( x 2 ⋅ l o g ( x ) ) d d x ( x x ) x = e x p ( x 2 ⋅ l o g ( x ) ) ⋅ ( 2 x ⋅ l o g ( x ) + x ) x a = e x p ( a ⋅ l o g ( x ) ) d d x x a = e x p ( a ⋅ l o g ( x ) ) ⋅ a x = a ⋅ x a − 1 x ( x a ) = e x p ( x a ⋅ log ( x ) ) d d x x ( x a ) = x ( x a ) ⋅ ( a ⋅ x a − 1 log ( x ) + x a − 1 ) = x ( x a ) x a − 1 ( a ⋅ log ( x ) + 1 ) d d x a x = a x log ( a ) x ( a x ) = exp ( a x log ( x ) ) d d x x ( a x ) = x ( a x ) ( a x log ( a ) log ( x ) + a x x ) a ( x x ) = exp ( x x log ( a ) ) d d x a ( x x ) = a ( x x ) ( e x p ( x ⋅ l o g ( x ) ) ⋅ ( 1 ⋅ l o g ( x ) + 1 ) ⋅ log ( a ) ) = a ( x x ) ⋅ x x ⋅ log ( a ) ⋅ ( 1 + log ( x ) ) ) {\displaystyle {\begin{matrix}a^{x}=\mathrm {exp} (x\cdot \mathrm {log} (a))\\x^{x}=\mathrm {exp} \left(x\cdot \mathrm {log} \left(x\right)\right)\\x^{(x^{x})}=\mathrm {exp} \left(x^{x}\mathrm {log} \left(x\right)\right)\\{\frac {d}{dx}}x^{x}=\mathrm {exp} \left(x\cdot \mathrm {log} \left(x\right)\right)\cdot \left(1\cdot \mathrm {log} \left(x\right)+1\right)\\{\frac {d}{dx}}x^{(x^{x})}=\mathrm {exp} \left(x^{x}\mathrm {log} \left(x\right)\right)\cdot \left(\left({\frac {d}{dx}}x^{x}\right)\mathrm {log} \left(x\right)+x^{x-1}\right)\\{\frac {d}{dx}}x^{(x^{x})}=\mathrm {exp} \left(x^{x}\mathrm {log} \left(x\right)\right)\cdot \left(\left(\mathrm {exp} \left(x\cdot \mathrm {log} \left(x\right)\right)\cdot \left(1\cdot \mathrm {log} \left(x\right)+1\right)\right)\mathrm {log} \left(x\right)+x^{x-1}\right)\\\\\left(x^{x}\right)^{x}=\mathrm {exp} \left(x\cdot \mathrm {log} \left(x^{x}\right)\right)=\mathrm {exp} \left(x\left(x\cdot \mathrm {log} \left(x\right)\right)\right)=\mathrm {exp} \left(x^{2}\cdot \mathrm {log} \left(x\right)\right)\\{\frac {d}{dx}}\left(x^{x}\right)^{x}=\mathrm {exp} \left(x^{2}\cdot \mathrm {log} \left(x\right)\right)\cdot \left(2x\cdot \mathrm {log} \left(x\right)+x\right)\\\\x^{a}=\mathrm {exp} \left(a\cdot \mathrm {log} \left(x\right)\right)\\{\frac {d}{dx}}x^{a}=\mathrm {exp} \left(a\cdot \mathrm {log} \left(x\right)\right)\cdot {\frac {a}{x}}=a\cdot x^{a-1}\\x^{\left(x^{a}\right)}=\mathrm {exp} \left(x^{a}\cdot \log \left(x\right)\right)\\{\frac {d}{dx}}x^{\left(x^{a}\right)}=x^{\left(x^{a}\right)}\cdot \left(a\cdot x^{a-1}\log \left(x\right)+x^{a-1}\right)=x^{\left(x^{a}\right)}x^{a-1}\left(a\cdot \log \left(x\right)+1\right)\\\\{\frac {d}{dx}}a^{x}=a^{x}\log \left(a\right)\\x^{\left(a^{x}\right)}=\exp \left(a^{x}\log \left(x\right)\right)\\{\frac {d}{dx}}x^{\left(a^{x}\right)}=x^{\left(a^{x}\right)}\left(a^{x}\log \left(a\right)\log \left(x\right)+{\frac {a^{x}}{x}}\right)\\\\a^{\left(x^{x}\right)}=\exp \left(x^{x}\log \left(a\right)\right)\\{\frac {d}{dx}}a^{\left(x^{x}\right)}=a^{\left(x^{x}\right)}\left(\mathrm {exp} \left(x\cdot \mathrm {log} \left(x\right)\right)\cdot \left(1\cdot \mathrm {log} \left(x\right)+1\right)\cdot \log \left(a\right)\right)=a^{\left(x^{x}\right)}\cdot x^{x}\cdot \log \left(a\right)\cdot \left(1+\log \left(x\right)\right))\end{matrix}}}
