1. **Problem:** Find the first derivative of $$f(x) = (x^5 - 2x + 4)(3x^2 - 4x - 6)$$ 2. **Formula:** Use the product rule for derivatives: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Step 1:** Identify $$u(x) = x^5 - 2x + 4$$ and $$v(x) = 3x^2 - 4x - 6$$ 4. **Step 2:** Compute derivatives: $$u'(x) = 5x^4 - 2$$ $$v'(x) = 6x - 4$$ 5. **Step 3:** Apply product rule: $$f'(x) = (5x^4 - 2)(3x^2 - 4x - 6) + (x^5 - 2x + 4)(6x - 4)$$ 6. **Step 4:** Expand both products: $$(5x^4)(3x^2) = 15x^6$$ $$(5x^4)(-4x) = -20x^5$$ $$(5x^4)(-6) = -30x^4$$ $$(-2)(3x^2) = -6x^2$$ $$(-2)(-4x) = 8x$$ $$(-2)(-6) = 12$$ $$(x^5)(6x) = 6x^6$$ $$(x^5)(-4) = -4x^5$$ $$(-2x)(6x) = -12x^2$$ $$(-2x)(-4) = 8x$$ $$(4)(6x) = 24x$$ $$(4)(-4) = -16$$ 7. **Step 5:** Combine all terms: $$f'(x) = 15x^6 - 20x^5 - 30x^4 - 6x^2 + 8x + 12 + 6x^6 - 4x^5 - 12x^2 + 8x + 24x - 16$$ 8. **Step 6:** Group like terms: $$f'(x) = (15x^6 + 6x^6) + (-20x^5 - 4x^5) + (-30x^4) + (-6x^2 - 12x^2) + (8x + 8x + 24x) + (12 - 16)$$ $$= 21x^6 - 24x^5 - 30x^4 - 18x^2 + 40x - 4$$ **Final answer:** $$\boxed{f'(x) = 21x^6 - 24x^5 - 30x^4 - 18x^2 + 40x - 4}$$