Given the function
$$f(x_1,x_2) = \min \{x_1,x_2 \}^\alpha$$
I have to find the partial derivatives for $x_1,x_2$ but I am just unsure whether it is as simple as I think it is or if it gets a bit tricky. I know that for $$ f(x_1,x_2) = \min \{x_1,x_2 \} = \begin{cases} x_1 \ & \text{if} \ x_1 \leq x_2 \\ x_2 & \text{if} \ x_1 > x_2 \end{cases} $$ Does this mean that for the given function we have $$ f(x_1,x_2) = \min \{x_1,x_2 \}^\alpha = \begin{cases} x_1^\alpha \ & \text{if} \ x_1 \leq x_2 \\ x_2^\alpha & \text{if} \ x_1 > x_2 \end{cases} $$ and thus the partial derivatives will be $$ \frac{ \partial f(x_1,x_2) }{ \partial x_1 } = \begin{cases} \alpha x_1^{\alpha-1} \ & \text{if} \ x_1 \leq x_2 \\ 0 & \text{if} \ x_1 > x_2 \end{cases} $$ and $$ \frac{ \partial f(x_1,x_2) }{ \partial x_2 } = \begin{cases} 0 \ & \text{if} \ x_1 \leq x_2 \\ \alpha x_2^{\alpha-1} & \text{if} \ x_1 > x_2 \end{cases} $$ Can you clarify? Thanks in advance.