Answer
$y'=\dfrac{1+\dfrac{1}{2x^{1/2}}}{3(x+x^{1/2})^{4/3}}=\dfrac{2x^{1/2}+1}{6x^{1/2}(x+x^{1/2})^{4/3}}$
Work Step by Step
$y=1/\sqrt[3]{x+\sqrt{x}}$
Start differentiating by using the quotient rule:
$y'=\dfrac{(\sqrt[3]{x+\sqrt{x}})(1)'-(1)(\sqrt[3]{x+\sqrt{x}})'}{(\sqrt[3]{x+\sqrt{x}})^{2}}=...$
$...=\dfrac{(\sqrt[3]{x+\sqrt{x}})(0)-(1)(\sqrt[3]{x+\sqrt{x}})'}{(\sqrt[3]{x+\sqrt{x}})^{2}}=\dfrac{-(\sqrt[3]{x+\sqrt{x}})'}{(\sqrt[3]{x+\sqrt{x}})^{2}}=...$
Rewrite $\sqrt[3]{x+\sqrt{x}}$ using rational exponents:
$...=\dfrac{-[(x+x^{1/2})^{1/3}]'}{[(x+x^{1/2})^{1/3}]^{2}}=\dfrac{-[(x+x^{1/2})^{1/3}]'}{(x+x^{1/2})^{2/3}}=...$
Use the chain rule to evaluate the derivative indicated in the numerator:
$...=\dfrac{-\Big[\dfrac{1}{3}(x+x^{1/2})^{-2/3}(x+x^{1/2})'\Big]}{(x+x^{1/2})^{2/3}}=...$
Evaluate the last derivative and simplify:
$...=-\dfrac{(x+x^{1/2})^{-2/3}\Big(1+\dfrac{1}{2}x^{-1/2}\Big)}{3(x+x^{1/2})^{2/3}}=\dfrac{1+\dfrac{1}{2x^{1/2}}}{3(x+x^{1/2})^{4/3}}=...$
$...=\dfrac{\dfrac{2x^{1/2}+1}{2x^{1/2}}}{3(x+x^{1/2})^{4/3}}=\dfrac{2x^{1/2}+1}{6x^{1/2}(x+x^{1/2})^{4/3}}$
