1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ of the function $y = \sqrt{x^2 + 2}$. 2. **Formula and rules:** Use the chain rule for derivatives. The derivative of $\sqrt{u}$ is $\frac{1}{2\sqrt{u}} \cdot \frac{du}{dx}$. Here, $u = x^2 + 2$. 3. **Intermediate work:** Calculate $\frac{du}{dx} = \frac{d}{dx}(x^2 + 2) = 2x$. 4. **Apply the chain rule:** $$\frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + 2}} \cdot 2x = \frac{2x}{2\sqrt{x^2 + 2}} = \frac{x}{\sqrt{x^2 + 2}}.$$ 5. **Explanation:** We first identified the inner function $u = x^2 + 2$, then found its derivative $2x$. Using the chain rule, we multiplied the derivative of the outer function $\sqrt{u}$ by the derivative of $u$. Simplifying gave the final derivative. **Final answer:** $$\frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 2}}.$$