7. (10%) Find the derivative of y with respect to x if y = 2( x)^x/2.

I. (10%) A stock portfolio is composed of two stocks: \( A_{1} \) and \( B_{1} \). Another stock portfolio also has two investments: stocks \( A_{2} \) and \( B_{2} \). Suppose the mean prices \( (\mu) \) and the standard deviations \( (\sigma) \) of \( A_{1} \) and \( A_{2} \) are equal, and the mean price and standard deviation of \( B_{1} \) are equal to those of \( B_{2} \) (i.e., \( \mu_{A_1} = \mu_{A_2}, \mu_{B_1} = \mu_{B_2}, \sigma_{A_1} = \sigma_{A_2}, \sigma_{B_1} = \sigma_{B_2} \)). Suppose each stock comprises 50% of the portfolio. However, the correlation between stocks is higher in the first portfolio (i.e., \( \rho_1 > \rho_2 \)). Explain the difference in expected returns and risks between the two portfolios.

7. (10%) Find the derivative of \( y \) with respect to \( x \) if \( y = 2(\ln x)^{x/2} \).

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