1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt{x} - 2$$. 2. **Recall the formula:** The derivative of $$x^n$$ with respect to $$x$$ is $$nx^{n-1}$$. 3. **Rewrite the function:** Express the square root as a power: $$f(x) = x^{\frac{1}{2}} - 2$$ 4. **Differentiate each term:** - Derivative of $$x^{\frac{1}{2}}$$ is $$\frac{1}{2}x^{\frac{1}{2} - 1} = \frac{1}{2}x^{-\frac{1}{2}}$$. - Derivative of constant $$-2$$ is $$0$$. 5. **Combine results:** $$f'(x) = \frac{1}{2}x^{-\frac{1}{2}} + 0 = \frac{1}{2}x^{-\frac{1}{2}}$$ 6. **Rewrite the derivative in radical form:** $$f'(x) = \frac{1}{2\sqrt{x}}$$ **Final answer:** $$\boxed{f'(x) = \frac{1}{2\sqrt{x}}}$$