1. **State the problem:** We need to find the first derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $$4x^2 - 2y^2 = 9.$$\n\n2. **Recall the formula and rules:** When differentiating implicitly, we differentiate both sides with respect to $x$. For terms involving $y$, we use the chain rule: $$\frac{d}{dx}[y^2] = 2y \frac{dy}{dx}.$$\n\n3. **Differentiate both sides:**\n$$\frac{d}{dx}[4x^2] - \frac{d}{dx}[2y^2] = \frac{d}{dx}[9]$$\n$$8x - 4y \frac{dy}{dx} = 0.$$\n\n4. **Isolate $\frac{dy}{dx}$:**\n$$8x = 4y \frac{dy}{dx}$$\nDivide both sides by $4y$:\n$$\frac{8x}{4y} = \frac{dy}{dx}$$\nShow cancellation:\n$$\frac{\cancel{8}2x}{\cancel{4}y} = \frac{dy}{dx}$$\n$$\frac{2x}{y} = \frac{dy}{dx}.$$\n\n5. **Final answer:**\n$$\boxed{\frac{dy}{dx} = \frac{2x}{y}}.$$