1. **State the problem:** Differentiate the function $y = (1 - x^2)(6x + 1)$ with respect to $x$. 2. **Recall the product rule:** If $y = uv$, then $$\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$$ where $u$ and $v$ are functions of $x$. 3. **Identify $u$ and $v$:** Here, $u = 1 - x^2$ and $v = 6x + 1$. 4. **Compute derivatives:** $$\frac{du}{dx} = \frac{d}{dx}(1 - x^2) = 0 - 2x = -2x$$ $$\frac{dv}{dx} = \frac{d}{dx}(6x + 1) = 6 + 0 = 6$$ 5. **Apply the product rule:** $$\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} = (1 - x^2)(6) + (6x + 1)(-2x)$$ 6. **Simplify each term:** $$(1 - x^2)(6) = 6 - 6x^2$$ $$(6x + 1)(-2x) = -12x^2 - 2x$$ 7. **Combine terms:** $$\frac{dy}{dx} = 6 - 6x^2 - 12x^2 - 2x = 6 - 18x^2 - 2x$$ 8. **Final answer:** $$\boxed{\frac{dy}{dx} = 6 - 18x^2 - 2x}$$