Burton Schofield

08/05/2024 · Elementary School

Find all the second partial derivatives \( z=\frac{x}{x+y} \) A. \( \frac{\partial^{2} z}{\partial x^{2}}=-\frac{2 y}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y^{2}}=\frac{2 x}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial^{2} z}{\partial x \partial y}=\frac{x-y}{(x+y)^{3}} \) B. \( \frac{\partial^{2} z}{\partial x^{2}}=\frac{2 y}{(x+y)^{3}(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y^{2}}=-\frac{2 x}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial^{2} z}{\partial x \partial y}=\frac{x-y}{(x+y)^{3}} \) C. \( \frac{\partial^{2} z}{\partial x^{2}}=-\frac{2 y}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y^{2}}=\frac{2 x}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial^{2} z}{\partial x \partial y}=\frac{y-x}{(x+y)^{3}} \) D. \( \frac{\partial^{2} z}{\partial x^{2}}=-\frac{y}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y^{2}}=\frac{x}{(x+y)^{3}} ; \frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial^{2} z}{\partial x \partial y}=\frac{x-y}{(x+y)^{3}} \)