1. Problem: Find the derivative of \(F(x) = 2x^2(3x^4 - 2)\). 2. Formula: Use the product rule for derivatives: \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\). 3. Let \(u(x) = 2x^2\) and \(v(x) = 3x^4 - 2\). 4. Compute derivatives: \(u'(x) = \frac{d}{dx}[2x^2] = 4x\) \(v'(x) = \frac{d}{dx}[3x^4 - 2] = 12x^3\) 5. Apply product rule: \(F'(x) = u'(x)v(x) + u(x)v'(x) = 4x(3x^4 - 2) + 2x^2(12x^3)\) 6. Simplify: \(4x(3x^4 - 2) = 12x^5 - 8x\) \(2x^2(12x^3) = 24x^5\) 7. Combine terms: \(F'(x) = 12x^5 - 8x + 24x^5 = (12x^5 + 24x^5) - 8x = 36x^5 - 8x\) Final answer: \(\boxed{F'(x) = 36x^5 - 8x}\)