Functions of two variables and partial derivatives practice question 11

The subscripts set the order: $f_{xy}$ means differentiate in $x$ first, then differentiate that result in $y$. Which rule does each stage need?

First $f_x =\dfrac{2x + y}{x^{2} + xy + 1}$ by the chain rule. Now differentiate that in $y$, holding $x$ fixed: it is a quotient.

In the quotient rule the numerator simplifies to $\left(x^{2} + xy + 1\right) - (2x + y)x = 1 - x^{2}$. For the check, $f_y = \dfrac{x}{x^{2} + xy + 1}$.