1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = 3x^3 + 4x^2 - 2x^{-1}$.\n\n2. **Recall the differentiation rules:**\n- Power rule: $\frac{d}{dx} x^n = nx^{n-1}$\n- Constant multiple rule: $\frac{d}{dx} [cf(x)] = c \frac{d}{dx} f(x)$\n\n3. **Differentiate each term:**\n- For $3x^3$, derivative is $3 \times 3x^{3-1} = 9x^2$\n- For $4x^2$, derivative is $4 \times 2x^{2-1} = 8x$\n- For $-2x^{-1}$, derivative is $-2 \times (-1)x^{-1-1} = 2x^{-2}$\n\n4. **Combine the results:**\n$$\frac{dy}{dx} = 9x^2 + 8x + 2x^{-2}$$\n\n5. **Final answer:**\n$$\boxed{\frac{dy}{dx} = 9x^2 + 8x + \frac{2}{x^2}}$$