1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = -12\sqrt{x^3 + 7x - 1}$. 2. **Recall the formula:** For $y = k\sqrt{u}$ where $u$ is a function of $x$, rewrite as $y = k u^{1/2}$. The derivative is given by the chain rule: $$\frac{dy}{dx} = k \cdot \frac{1}{2} u^{-1/2} \cdot \frac{du}{dx}$$ 3. **Identify $u$ and $k$:** Here, $k = -12$ and $u = x^3 + 7x - 1$. 4. **Compute $\frac{du}{dx}$:** $$\frac{du}{dx} = 3x^2 + 7$$ 5. **Apply the chain rule:** $$\frac{dy}{dx} = -12 \cdot \frac{1}{2} (x^3 + 7x - 1)^{-1/2} (3x^2 + 7)$$ 6. **Simplify constants:** $$\frac{dy}{dx} = -6 (x^3 + 7x - 1)^{-1/2} (3x^2 + 7)$$ 7. **Rewrite in radical form:** $$\frac{dy}{dx} = \frac{-6 (3x^2 + 7)}{\sqrt{x^3 + 7x - 1}}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{-6 (3x^2 + 7)}{\sqrt{x^3 + 7x - 1}}}$$