Answer
$$y' = \frac{1}{{3\left( {x + 1} \right)}}$$
Work Step by Step
$$\eqalign{
& y = \ln \left( {\root 3 \of {x + 1} } \right) \cr
& or \cr
& y = \ln {\left( {x + 1} \right)^{1/3}} \cr
& {\text{logarithmic property }}\ln {u^v} = v\ln u \cr
& y = \frac{1}{3}\ln \left( {x + 1} \right) \cr
& {\text{find the derivative}} \cr
& y' = \left( {\frac{1}{3}\ln \left( {x + 1} \right)} \right)' \cr
& y' = \frac{1}{3}\left( {\ln \left( {x + 1} \right)} \right)' \cr
& {\text{use }}\left( {\ln u} \right)' = \frac{{u'}}{u}, \cr
& y' = \frac{1}{3}\left( {\frac{{\left( {x + 1} \right)'}}{{x + 1}}} \right) \cr
& y' = \frac{1}{3}\left( {\frac{1}{{x + 1}}} \right) \cr
& {\text{simplifying}} \cr
& y' = \frac{1}{{3\left( {x + 1} \right)}} \cr} $$
