1. **Problem:** Find the derivative $\frac{dy}{dx}$ if $y = \frac{x^2 + 1}{\sqrt{x}}$. 2. **Formula and rules:** Use the quotient rule or rewrite the function to simplify differentiation. 3. Rewrite $y$ as: $$y = \frac{x^2 + 1}{x^{1/2}} = (x^2 + 1) x^{-1/2} = x^{2 - \frac{1}{2}} + x^{0 - \frac{1}{2}} = x^{3/2} + x^{-1/2}$$ 4. Differentiate term-by-term using the power rule $\frac{d}{dx} x^n = n x^{n-1}$: $$\frac{dy}{dx} = \frac{d}{dx} x^{3/2} + \frac{d}{dx} x^{-1/2} = \frac{3}{2} x^{\frac{3}{2} - 1} - \frac{1}{2} x^{-\frac{1}{2} - 1} = \frac{3}{2} x^{1/2} - \frac{1}{2} x^{-3/2}$$ 5. **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{3}{2} \sqrt{x} - \frac{1}{2} \frac{1}{x^{3/2}}}$$