1. **State the problem:** Differentiate the function $$y = 2x \log_3(\sqrt{x})$$. 2. **Recall the formula and rules:** - The derivative of a product $$uv$$ is $$u'v + uv'$$. - Change of base formula for logarithms: $$\log_a b = \frac{\ln b}{\ln a}$$. - Derivative of $$\ln x$$ is $$\frac{1}{x}$$. - Derivative of $$x^n$$ is $$nx^{n-1}$$. 3. **Rewrite the function:** $$y = 2x \log_3(x^{1/2}) = 2x \cdot \frac{\ln(x^{1/2})}{\ln 3} = 2x \cdot \frac{1}{2} \frac{\ln x}{\ln 3} = \frac{2x}{2} \cdot \frac{\ln x}{\ln 3} = \frac{x \ln x}{\ln 3}$$ 4. **Differentiate using product rule:** Let $$u = x$$ and $$v = \frac{\ln x}{\ln 3}$$. $$u' = 1$$ $$v' = \frac{1}{x \ln 3}$$ Then, $$y' = u'v + uv' = 1 \cdot \frac{\ln x}{\ln 3} + x \cdot \frac{1}{x \ln 3} = \frac{\ln x}{\ln 3} + \frac{1}{\ln 3}$$ 5. **Final answer:** $$y' = \frac{\ln x + 1}{\ln 3}$$