\begin{matrix} f_{x}^{′} (x y) & = & (sin x \cdot cos y) & product \\ = & (sin x \cdot cos y) & trig \\ = & ((cos x \cdot x) \cdot cos y) & coefficient \\ = & ((cos x \cdot 1) \cdot cos y) & identity \\ = & (cos x \cdot cos y) \\ f_{y}^{′} (x y) & = & (cos x \cdot cos y) & product \\ = & (cos x \cdot cos y) & trig \\ = & (cos x \cdot (- sin y \cdot y)) & coefficient \\ = & (cos x \cdot (- sin y \cdot 1)) & identity \\ = & (cos x \cdot - sin y) \end{matrix}