Differentiate f(x)=(x+1)ˣ⁺¹

The question below was obtained from one of the past questions for the Skoltech institute entrance examination for admission into computational science masters program. Skoltech is a reputable and prestigious institute in Moscow Russia that offers scholarships to international students. 

If $f(x)=(x+1)^{x+1}$, find $f'(x)$.

$f(x)=(x+1)^{x+1}$

Take the logarithm of both sides 

$\ln{f(x)}=(x+1)\ln(x+1)$

Differentiate both sides 

$\frac{1}{f(x)}f'(x)=\ln(x+1)+(x+1)\frac{1}{x+1}$

$\frac{1}{f(x)}f'(x)=\ln(x+1)+1$

$f'(x)=f(x)[\ln(x+1)+1]$

$=(x+1)^{x+1}[\ln(x+1)+1]$