Question

find the derivative:

$f(x)=\sqrt3{x}$

a. $\frac{1}{3}$

b. $\frac{1}{3}x$

c. $\frac{1}{3}x^{1/3}$

d. $\frac{1}{3}x^{-2/3}$

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\sqrt[3]{x} \) as \( f(x)=x^{\frac{1}{3}} \).

Step2: Apply the power rule

The power rule for differentiation is \( (x^n)^\prime = nx^{n - 1} \).
For \( n=\frac{1}{3} \), we have \( f^\prime(x)=\frac{1}{3}x^{\frac{1}{3}-1}=\frac{1}{3}x^{-\frac{2}{3}} \).

Answer:

D. \( \frac{1}{3}x^{-\frac{2}{3}} \)