Question

Для функции z=ln(u^2+v^2),где u=x cos y v=y sin x найти частные производные dz/dx и dz/dy

Answer 1

$z=ln(u^2+v^2)\; ,\; \; u=x\, cosy\; \; ,\; \; v=y\, sinx\\\\ \frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} +\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial x} = \frac{2u}{u^2+v^2} \cdot cosy+\frac{2v}{u^2+v^2}\cdot y\, cosx \\\\ \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial y} = \frac{2u}{u^2+v^2}\cdot (-xsiny)+\frac{2v}{u^2+v^2}\cdot sinx$