find the derivative:\n\n$f(x)=\\sqrt3{x}$\n\na. $\\frac{1}{3}$\n\nb. $\\frac{1}{3}x$\n\nc…

find the derivative:\n\n$f(x)=\\sqrt3{x}$\n\na. $\\frac{1}{3}$\n\nb. $\\frac{1}{3}x$\n\nc. $\\frac{1}{3}x^{1/3}$\n\nd. $\\frac{1}{3}x^{-2/3}$

find the derivative:\n\n$f(x)=\\sqrt3{x}$\n\na. $\\frac{1}{3}$\n\nb. $\\frac{1}{3}x$\n\nc. $\\frac{1}{3}x^{1/3}$\n\nd. $\\frac{1}{3}x^{-2/3}$

Answer

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}} )