Question

let $f(x)=\sqrt{x}$. $f(x)=$

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\sqrt{x} \) as \( f(x)=x^{\frac{1}{2}} \).

Step2: Apply the power rule

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

Step3: Simplify the exponent

\( \frac{1}{2}-1=-\frac{1}{2} \), so \( f^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}} \).

Step4: Rewrite with positive exponent

\( x^{-\frac{1}{2}}=\frac{1}{x^{\frac{1}{2}}}=\frac{1}{\sqrt{x}} \), then \( f^\prime(x)=\frac{1}{2\sqrt{x}} \).

Answer:

\( \frac{1}{2\sqrt{x}} \)