1. **State the problem:** Differentiate the function $$y(x) = 2x\sqrt{x} - \sqrt{5}x + \frac{2}{3}\sqrt{x} - \frac{5}{3}x - \frac{1}{2}x\sqrt{3x} - 1$$ with respect to $x$. 2. **Rewrite the function using exponents:** Recall that $\sqrt{x} = x^{\frac{1}{2}}$ and $\sqrt{3x} = \sqrt{3} \sqrt{x} = \sqrt{3} x^{\frac{1}{2}}$. So, $$y = 2x x^{\frac{1}{2}} - \sqrt{5} x + \frac{2}{3} x^{\frac{1}{2}} - \frac{5}{3} x - \frac{1}{2} x \sqrt{3} x^{\frac{1}{2}} - 1$$ Simplify terms: $$y = 2 x^{\frac{3}{2}} - \sqrt{5} x + \frac{2}{3} x^{\frac{1}{2}} - \frac{5}{3} x - \frac{1}{2} \sqrt{3} x^{\frac{3}{2}} - 1$$ 3. **Differentiate term-by-term using the power rule:** For $x^n$, $\frac{d}{dx} x^n = n x^{n-1}$. - $\frac{d}{dx} 2 x^{\frac{3}{2}} = 2 \cdot \frac{3}{2} x^{\frac{3}{2} - 1} = 3 x^{\frac{1}{2}}$ - $\frac{d}{dx} (- \sqrt{5} x) = - \sqrt{5}$ - $\frac{d}{dx} \frac{2}{3} x^{\frac{1}{2}} = \frac{2}{3} \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{1}{3} x^{-\frac{1}{2}}$ - $\frac{d}{dx} (- \frac{5}{3} x) = - \frac{5}{3}$ - $\frac{d}{dx} \left(- \frac{1}{2} \sqrt{3} x^{\frac{3}{2}} \right) = - \frac{1}{2} \sqrt{3} \cdot \frac{3}{2} x^{\frac{3}{2} - 1} = - \frac{3}{4} \sqrt{3} x^{\frac{1}{2}}$ - $\frac{d}{dx} (-1) = 0$ 4. **Combine all derivatives:** $$y'(x) = 3 x^{\frac{1}{2}} - \sqrt{5} + \frac{1}{3} x^{-\frac{1}{2}} - \frac{5}{3} - \frac{3}{4} \sqrt{3} x^{\frac{1}{2}}$$ 5. **Rewrite in radical form:** $$y'(x) = 3 \sqrt{x} - \sqrt{5} + \frac{1}{3 \sqrt{x}} - \frac{5}{3} - \frac{3}{4} \sqrt{3} \sqrt{x}$$ This is the derivative of the given function.