From the image provided, the mathematical expressions seem to involve a utility function $U$, defined as:

$U = \frac{e_2}{e_1 + e_2}(100 - C e_2),$

and its derivative with respect to $e_2$:

$\frac{\partial U}{\partial e_2} = \frac{e_1}{(e_1 + e_2)^2}(100 - C e_2) - \frac{e_2 C}{e_1 + e_2}.$

The condition set by this derivative is $\frac{\partial U}{\partial e_2} = 0$, which implies solving the equation for critical points of $U$ with respect to $e_2$.

Steps for Analysis:

1. Expression Breakdown:

   • The utility function $U$ combines proportions (via $\frac{e_2}{e_1 + e_2}$) with the linear term $100 - C e_2$.
   • The derivative combines a quotient rule application with the simplification of terms.

2. Critical Points: Solving $\frac{\partial U}{\partial e_2} = 0$ involves:

   • Equating the numerator of the derivative's simplified form to 0.