Exercise
Find the derivative of the following function:
f(x)=\sqrt[3]{x+\sqrt{x}}
Final Answer
Solution
First, we simplify the function:
f(x)={(x+\sqrt{x})}^{\frac{1}{3}}
Using Derivative formulas, we get the derivative:
f'(x)=\frac{1}{3}{(x+\sqrt{x})}^{-\frac{2}{3}}\cdot (1+ \frac{1}{2\sqrt{x}})=
=\frac{1}{3\sqrt[3]{{(x+\sqrt{x})}^2}}\cdot \frac{2\sqrt{x}+1}{2\sqrt{x}}=
=\frac{2\sqrt{x}+1}{6\sqrt{x}\sqrt[3]{(x+\sqrt{x})}^2}
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