# Partial Derivative wrt x of sin x y over cos (x + y)

## Theorem

$\dfrac \partial {\partial x} \dfrac {\sin x y} {\map \cos {x + y} } = \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$

## Proof

 $\ds \frac \partial {\partial x} \frac {\sin x y} {\map \cos {x + y} }$ $=$ $\ds \frac {\map \cos {x + y} y \cos x y - \sin x y \paren {-\map \sin {x + y} } } {\map {\cos^2} {x + y} }$ Quotient Rule for Derivatives, treating $y$ as a constant $\ds$ $=$ $\ds \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$ simplifying

$\blacksquare$