Question

if ( f ) is the function defined by ( f(x)=sqrt4{x} ), what is ( f^{prime}(x) )?
a ( \frac{1}{4} x^{\frac{1}{4}} )
b ( x^{-\frac{3}{4}} )
c ( \frac{1}{4} x^{-\frac{3}{4}} )
d ( 4 cdot sqrt3{x} )

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