Mathematics

The key is recognizing that the argument of $\tan^{-1}$ has the form of a tangent subtraction formula. If you rewrite the fraction as $\tan(\alpha - \beta)$ for carefully chosen angles $\alpha$ and $\beta$, the inverse tangent will collapse to a simple linear expression in $x$. Once $y$ simplifies, differentiation becomes straightforward.

The key is recognizing that the argument of $\tan^{-1}$ has the form of a tangent subtraction formula. If you rewrite the fraction as $\tan(\alpha - \beta)$ for carefully chosen angles $\alpha$ and $\beta$, the inverse tangent will collapse to a simple linear expression in $x$. Once $y$ simplifies, differentiation becomes straightforward.