Derivatives of functions with exponents

Determine the derivatives of the functions

Variant 1
\( e^{\sqrt{x+1}} \)
Result
\(\displaystyle f'=\frac{e^{\sqrt{x+1}}}{2\sqrt{x+1}} \), \(D_f=D_{f'}=\langle-1,\infty)\)
Variant 2
\(xe^{\frac1x}+e^{-\sqrt x}\)
Variant 3
\( x^x\)
Result
\(\displaystyle f'=x^x(\ln x+1) \), \(D_f=D_{f'}=\langle0,\infty)\)
Variant 4
\( x^{\sqrt x}\)
Result
\(\displaystyle f'=x^{\sqrt x}\frac{2-\ln x}{2\sqrt x} \), \(D_f=\langle0,\infty)\), \(D_{f'}=(0,\infty)\)
Variant 5
\( x^{\frac1x}\)
Result
\(\displaystyle f'=x^{\frac1x}\frac{1-\ln x}{x^2} \), \(D_f=D_{f'}=(0,\infty)\)
Variant 6
\( x^{\ln x}\)
Result
\(\displaystyle f'=\frac{2x^{\ln x}\ln x}x \), \(D_f=D_{f'}=(0,\infty)\)
Variant 7
\(\displaystyle \left(\frac{x}{x+1}\right)^x \)
Variant 8
\( (x^2+1)^{\sin x} \)
Result
\(\displaystyle f'=(x^2+1)^{\sin x}\left(\cos x\ln(x^2+1)-\frac{2\sin x}{x^2+1}\right)\), \(D_f=D_{f'}=\mathbb R\)
Variant 9
\( (\sin x)^{\cos x} \)