Question

U(x1​,x2​)=$(\sqrt{x_1}+\sqrt{x_2})^2$

Answer 1

Find the partial derivatives of X1 and X2 using the utility function

U($X_{1}$ ​,$X_{2}$ ​)= $( x_1^{0.5} + x_2^{0.5} )^{2}$

Using chain rule we partially differentiate the utility function with respect to $X_{1} and X_{2}$

Let Z = $x_1^{0.5} + x_2^{0.5}$

$\frac{ɗz}{ɗx_{1}} = \frac{0.5}{X_{1}^{0.5}}$

$\frac{ɗz}{ɗx_{2}} = \frac{0.5}{X_{2}^{0.5}}$

$U(X_{1} ​ ​,X_{2} ​ ​)$ = Z$^{2}$

$\frac{ɗu}{ɗz} =$ 2Z

$\frac{ɗu}{ɗx_{1}} = \frac{0.5}{X_{1}^{0.5}} \times 2[x_1^{0.5} + x_2^{0.5}]$

$\frac{ɗu}{ɗx_{1}} = 1 + 2x_2^{0.5}$

$\frac{ɗu}{ɗx_{2}} = \frac{0.5}{X_{2}^{0.5}} \times 2[x_1^{0.5} + x_2^{0.5}]$

$\frac{ɗu}{ɗx_{2}} = 1 + 2x_1^{0.5}$