if ( f ) is the function defined by ( f(x)=sqrt4{x} ), what is ( f^{prime}(x) )?\na ( \frac{1}{4}…

if ( f ) is the function defined by ( f(x)=sqrt4{x} ), what is ( f^{prime}(x) )?\na ( \frac{1}{4} x^{\frac{1}{4}} )\nb ( x^{-\frac{3}{4}} )\nc ( \frac{1}{4} x^{-\frac{3}{4}} )\nd ( 4 cdot sqrt3{x} )

if ( f ) is the function defined by ( f(x)=sqrt4{x} ), what is ( f^{prime}(x) )?\na ( \frac{1}{4} x^{\frac{1}{4}} )\nb ( x^{-\frac{3}{4}} )\nc ( \frac{1}{4} x^{-\frac{3}{4}} )\nd ( 4 cdot sqrt3{x} )

Answer

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\sqrt[4]{x} ) as ( f(x)=x^{\frac{1}{4}} ) using the rule ( \sqrt[n]{a}=a^{\frac{1}{n}} ).

Step2: Apply the power rule

The power rule for differentiation is ( \frac{d}{dx}(x^n)=nx^{n - 1} ). For ( f(x)=x^{\frac{1}{4}} ), we have ( n=\frac{1}{4} ). Then ( f^{\prime}(x)=\frac{1}{4}x^{\frac{1}{4}-1}=\frac{1}{4}x^{-\frac{3}{4}} ).

Answer:

C. ( \frac{1}{4}x^{-\frac{3}{4}} )