Answer
$y'=\dfrac{4\lambda(x+\lambda)^{3}(\lambda^{3}-x^{3})}{(x^{4}+\lambda^{4})^{2}}$
Work Step by Step
$y=\dfrac{(x+\lambda)^{4}}{x^{4}+\lambda^{4}}$
Start the differentiation process by using the quotient rule:
$y'=\dfrac{(x^{4}+\lambda^{4})[(x+\lambda)^{4}]'-(x+\lambda)^{4}(x^{4}+\lambda^{4})'}{(x^{4}+\lambda^{4})^{2}}=...$
Use the chain rule to evaluate the derivatives indicated in the numerator:
$...=\dfrac{(x^{4}+\lambda^{4})[4(x+\lambda)^{3}(1)]-(x+\lambda)^{4}(4x^{3})}{(x^{4}+\lambda^{4})^{2}}=...$
$...=\dfrac{4(x^{4}+\lambda^{4})(x+\lambda)^{3}-4x^{3}(x+\lambda)^{4}}{(x^{4}+\lambda^{4})^{2}}=...$
Take out common factor $4(x+\lambda)^{3}$ from the numerator and simplify:
$...=\dfrac{4(x+\lambda)^{3}[x^{4}+\lambda^{4}-x^{3}(x+\lambda)]}{(x^{4}+\lambda^{4})^{2}}=...$
$...=\dfrac{4(x+\lambda)^{3}(x^{4}+\lambda^{4}-x^{4}-x^{3}\lambda)}{(x^{4}+\lambda^{4})^{2}}=\dfrac{4(x+\lambda)^{3}(\lambda^{4}-x^{3}\lambda)}{(x^{4}+\lambda^{4})^{2}}=...$
Take common factor $\lambda$ from the numerator:
$...=\dfrac{4\lambda(x+\lambda)^{3}(\lambda^{3}-x^{3})}{(x^{4}+\lambda^{4})^{2}}$
