1. **State the problem:** Differentiate the function $y = x^2$ with respect to $x$. 2. **Recall the differentiation rule:** The power rule states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. 3. **Apply the power rule:** Here, $n = 2$, so $$\frac{dy}{dx} = 2x^{2-1} = 2x.$$ 4. **Interpretation:** The derivative $2x$ represents the slope of the tangent line to the curve $y = x^2$ at any point $x$. **Final answer:** $$\frac{dy}{dx} = 2x.$$