1. **State the problem:** Find the partial derivative of the function $$f(x,y) = e^{xy} \cos(x) \sin(y)$$ with respect to $$y$$, denoted as $$f_y(x,y)$$. 2. **Recall the formula and rules:** To find $$f_y$$, treat $$x$$ as a constant and differentiate with respect to $$y$$. Use the product rule: $$\frac{d}{dy}[u(y)v(y)] = u'(y)v(y) + u(y)v'(y)$$. 3. **Identify parts:** Let $$u(y) = e^{xy}$$ and $$v(y) = \cos(x) \sin(y)$$. Note $$\cos(x)$$ is constant with respect to $$y$$. 4. **Differentiate $$u(y)$$:** $$\frac{d}{dy} e^{xy} = e^{xy} \cdot \frac{d}{dy}(xy) = e^{xy} \cdot x$$. 5. **Differentiate $$v(y)$$:** $$\frac{d}{dy} [\cos(x) \sin(y)] = \cos(x) \cdot \cos(y)$$. 6. **Apply product rule:** $$ f_y = u'(y)v(y) + u(y)v'(y) = (e^{xy} x)(\cos(x) \sin(y)) + (e^{xy})(\cos(x) \cos(y)) $$ 7. **Factor common terms:** $$ f_y = e^{xy} \cos(x) (x \sin(y) + \cos(y)) $$ **Final answer:** $$ f_y(x,y) = e^{xy} \cos(x) (x \sin(y) + \cos(y)) $$