Functions of two variables and partial derivatives practice question 9

With $x$ held fixed, $f$ is arctan of an expression in $y$. What is the inside, and which rule handles an outside function and an inside one?

Hold $x$ fixed and use the chain rule with $\dfrac{d}{du}\arctan u = \dfrac{1}{1 + u^{2}}$ and $u = \dfrac{y}{x}$.

With $x$ fixed, $\dfrac{y}{x}$ is $y$ times the constant $\dfrac{1}{x}$, so its $y$-derivative is $\dfrac{1}{x}$. Then multiply the top and bottom by $x^{2}$ to simplify.