1. **Problem statement:** Find the derivative of the function $y = f(x) = 2x^2$. 2. **Formula used:** The derivative of a function $f(x)$ with respect to $x$ is given by $f'(x) = \frac{d}{dx}f(x)$. 3. **Important rule:** The power rule for derivatives states that $\frac{d}{dx} x^n = nx^{n-1}$ where $n$ is a constant. 4. **Apply the power rule:** For $y = 2x^2$, the constant 2 remains as a coefficient, and the derivative of $x^2$ is $2x^{2-1} = 2x$. 5. **Calculate the derivative:** $$ \frac{dy}{dx} = 2 \times 2x = 4x $$ 6. **Final answer:** The derivative of $y = 2x^2$ is $$ \boxed{\frac{dy}{dx} = 4x} $$