Let w=f(u,v)w = f(u, v)w=f(u,v) where u=xyu = xyu=xy and v=yxv = \frac{y}{x}v=xy. Which expression correctly gives ∂w∂x\frac{\partial w}{\partial x}∂x∂w by the chain rule?
∂w∂x=∂f∂u⋅y+∂f∂v⋅yx2\frac{\partial w}{\partial x} = \frac{\partial f}{\partial u} \cdot y + \frac{\partial f}{\partial v} \cdot \frac{y}{x^2}∂x∂w=∂u∂f⋅y+∂v∂f⋅x2y
∂w∂x=∂f∂u⋅y−∂f∂v⋅yx2\frac{\partial w}{\partial x} = \frac{\partial f}{\partial u} \cdot y - \frac{\partial f}{\partial v} \cdot \frac{y}{x^2}∂x∂w=∂u∂f⋅y−∂v∂f⋅x2y
∂w∂x=∂f∂u⋅y+∂f∂v⋅(−yx2)\frac{\partial w}{\partial x} = \frac{\partial f}{\partial u} \cdot y + \frac{\partial f}{\partial v} \cdot \left(-\frac{y}{x^2}\right)∂x∂w=∂u∂f⋅y+∂v∂f⋅(−x2y)
∂w∂x=∂f∂u⋅1y+∂f∂v⋅x\frac{\partial w}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{1}{y} + \frac{\partial f}{\partial v} \cdot x∂x∂w=∂u∂f⋅y1+∂v∂f⋅x