1. The problem asks to find the derivative of the function $f(x) = \sqrt{x}$ with respect to $x$. 2. Recall that $\sqrt{x} = x^{\frac{1}{2}}$. 3. The power rule for derivatives states that if $f(x) = x^n$, then $f'(x) = n x^{n-1}$. 4. Applying the power rule here: $$\frac{d}{dx} x^{\frac{1}{2}} = \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{1}{2} x^{-\frac{1}{2}}$$ 5. Rewrite $x^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{x}}$: $$\frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2 \sqrt{x}}$$ 6. Therefore, the derivative is: $$\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}$$ 7. Among the options given, option 3 matches this result. Final answer: Option 3: $\frac{1}{2 \sqrt{x}}$