1. **State the problem:** Find the first-order derivative of the function $$f(x) = (x+3)(x^3+2)$$. 2. **Recall the product rule:** For two functions $$u(x)$$ and $$v(x)$$, the derivative of their product is $$f'(x) = u'(x)v(x) + u(x)v'(x)$$. 3. **Identify the parts:** Let $$u(x) = x+3$$ and $$v(x) = x^3+2$$. 4. **Find the derivatives:** - $$u'(x) = 1$$ because the derivative of $$x$$ is 1 and the derivative of a constant is 0. - $$v'(x) = 3x^2$$ because the derivative of $$x^3$$ is $$3x^2$$ and the derivative of a constant is 0. 5. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 1 \cdot (x^3+2) + (x+3) \cdot 3x^2$$ 6. **Simplify:** $$f'(x) = x^3 + 2 + 3x^2(x+3)$$ 7. **Distribute:** $$f'(x) = x^3 + 2 + 3x^3 + 9x^2$$ 8. **Combine like terms:** $$f'(x) = (x^3 + 3x^3) + 9x^2 + 2 = 4x^3 + 9x^2 + 2$$ **Final answer:** $$f'(x) = 4x^3 + 9x^2 + 2$$