Question

求f(x/y)的一阶偏导数

Answer 1

问：求f(x/y)的一阶偏导数

答：亲，你好！为您找寻的答案：设 $u=\frac{x}{y}$，则 $f(x/y)=f(u)$。根据链式法则，有：$$\frac{\partial f(x/y)}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} = \frac{1}{y} \cdot \frac{\partial f}{\partial u}$$$$\frac{\partial f(x/y)}{\partial y} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y} = \frac{-x}{y^2} \cdot \frac{\partial f}{\partial u}$$因此，一阶偏导数为：$$\frac{\partial f(x/y)}{\partial x} = \frac{1}{y} \cdot \frac{\partial f}{\partial u}, \quad \frac{\partial f(x/y)}{\partial y} = \frac{-x}{y^2} \cdot \frac{\partial f}{\partial u}