Derivative Product 48124F

1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = (2 - x - 3x^3)(7 + x^5)$$. 2. **Formula used:** To differentiate a product of two functions, use the product rule: $$f'(x) = u'(x)v(x) + u(x)v'(x)$$ where $$u(x) = 2 - x - 3x^3$$ and $$v(x) = 7 + x^5$$. 3. **Find derivatives of each part:** - $$u'(x) = \frac{d}{dx}(2) - \frac{d}{dx}(x) - \frac{d}{dx}(3x^3) = 0 - 1 - 9x^2 = -1 - 9x^2$$ - $$v'(x) = \frac{d}{dx}(7) + \frac{d}{dx}(x^5) = 0 + 5x^4 = 5x^4$$ 4. **Apply the product rule:** $$f'(x) = (-1 - 9x^2)(7 + x^5) + (2 - x - 3x^3)(5x^4)$$ 5. **Expand each product:** $$(-1)(7) + (-1)(x^5) + (-9x^2)(7) + (-9x^2)(x^5) + (2)(5x^4) + (-x)(5x^4) + (-3x^3)(5x^4)$$ 6. **Simplify terms:** $$-7 - x^5 - 63x^2 - 9x^{7} + 10x^4 - 5x^5 - 15x^{7}$$ 7. **Combine like terms:** - Combine $$- x^5$$ and $$- 5x^5$$ to get $$-6x^5$$ - Combine $$- 9x^{7}$$ and $$- 15x^{7}$$ to get $$-24x^{7}$$ Final derivative: $$f'(x) = -7 - 63x^2 + 10x^4 - 6x^5 - 24x^{7}$$