Use logarithmic differention to find the dervative of y with respect to the given independent variable

y  = (x + 1)^x        take the ln of both sides

ln y  =  ln (x + 1)^x

ln y  =  x ln (x + 1)       differentiate both sides

y' / y  =  ln (x + 1)  + x (1/ (x + 1) )

y'  = y [ ln (x + 1)  + x/(x + 1) ]

y ' =  (x + 1)^x  [ ln (x + 1)  +  x /(x + 1) ]