1. **Stating the problem:** Differentiate the function $$f(x) = \frac{3x^2 - x}{\sqrt{1-2x}}$$ using the quotient rule. 2. **Recall the quotient rule:** For $$f(x) = \frac{u(x)}{v(x)}$$, the derivative is $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$$ 3. **Identify functions:** $$u(x) = 3x^2 - x$$ $$v(x) = \sqrt{1-2x} = (1-2x)^{1/2}$$ 4. **Compute derivatives:** $$u'(x) = 6x - 1$$ $$v'(x) = \frac{1}{2}(1-2x)^{-1/2} \cdot (-2) = -\frac{1}{\sqrt{1-2x}}$$ 5. **Apply quotient rule:** $$f'(x) = \frac{(6x - 1)\sqrt{1-2x} - (3x^2 - x)\left(-\frac{1}{\sqrt{1-2x}}\right)}{(\sqrt{1-2x})^2}$$ 6. **Simplify numerator:** $$= \frac{(6x - 1)\sqrt{1-2x} + \frac{3x^2 - x}{\sqrt{1-2x}}}{1-2x}$$ 7. **Combine terms over common denominator $$\sqrt{1-2x}$$:** $$= \frac{(6x - 1)(1-2x) + (3x^2 - x)}{(1-2x)\sqrt{1-2x}}$$ 8. **Expand numerator:** $$(6x - 1)(1-2x) = 6x - 12x^2 - 1 + 2x = -12x^2 + 8x - 1$$ 9. **Add remaining term:** $$-12x^2 + 8x - 1 + 3x^2 - x = -9x^2 + 7x - 1$$ 10. **Final derivative:** $$f'(x) = \frac{-9x^2 + 7x - 1}{(1-2x)\sqrt{1-2x}}$$ This is the derivative of the given function.