Answer
$4\dfrac {\left( 2\left( u+1\right) -u^{2}\right) \left( u-1\right) ^{3}}{\left( u^{2}+u+1\right) ^{5}}$
Work Step by Step
$\dfrac {d}{dx}\left( \dfrac {u-1}{u^{2}+u+1}\right) ^{4}=4\times \left( \dfrac {u-1}{u^{2}+u+1}\right) ^{3}\times \dfrac {d}{dx}\left( \dfrac {u-1}{u^{2}+u+1}\right) =4\left( \dfrac {u-1}{u^{2}+u+1}\right) ^{3}\times \dfrac {1\times \left( u^{2}+u+1\right) -\left( 2u+1\right) \left( u-1\right) }{\left( u^{2}+u+1\right) 2}=4\dfrac {\left( 2\left( u+1\right) -u^{2}\right) \left( u-1\right) ^{3}}{\left( u^{2}+u+1\right) ^{5}}$
