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Given the function $f(x)=\frac{2 \sqrt{x}}{3}$, find $f^{\prime}(x)$. Express your answer in radical form without using negative exponents, simplifying all fractions.
Answer Attempt 1 out of 2
\[
f^{\prime}(x)=
\]
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Answer
So, the derivative of the function \(f(x)=\frac{2 \sqrt{x}}{3}\) is \(\boxed{f^{\prime}(x)=\frac{1}{3\sqrt{x}}}\)
Step 1 ：Rewrite the function in a form that allows us to apply the power rule. The square root of \(x\) is \(x^{1/2}\), so we have: \(f(x)=\frac{2x^{1/2}}{3}\)
Step 2 ：Apply the power rule. The derivative of \(x^{1/2}\) is \(\frac{1}{2}x^{-1/2}\), so: \(f^{\prime}(x)=\frac{2}{3} \cdot \frac{1}{2}x^{-1/2}\)
Step 3 ：Simplify the constants: \(f^{\prime}(x)=\frac{1}{3}x^{-1/2}\)
Step 4 ：Rewrite the answer in radical form without using negative exponents: \(f^{\prime}(x)=\frac{1}{3\sqrt{x}}\)
Step 5 ：So, the derivative of the function \(f(x)=\frac{2 \sqrt{x}}{3}\) is \(\boxed{f^{\prime}(x)=\frac{1}{3\sqrt{x}}}\)
