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