Question

Найти полный дифференциал 2 порядка.

Answer 1

$z=9x^{-3}+2x^2y^4-3y \cos x \\ d^2z= \frac{\partial^2 z}{\partial x}^2dx^2 +2 \frac{\partial^2 z}{\partial x \partial y} dxdy+ \frac{\partial^2 z}{\partial y}^2dy^2\\ \frac{\partial z}{\partial x} = -27x^{-4}+4xy^4+3y \sin x\\ \frac{\partial z}{\partial y} = 8x^{2}y^3-3 \cos x\\ \frac{\partial^2 z}{\partial x^2} = 108x^{-5}+4y^4+3y \cos x \\ \frac{\partial^2 z}{\partial y^2} = 24x^{2}y^2$

$\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x}(\frac{\partial z}{\partial y}) = \frac{\partial}{\partial y}(\frac{\partial z}{\partial x}) = 16xy^3+3 \sin x$

$d^2z=(108x^{-5}+4y^4+3y \cos x)dx^2+2(16xy^3+3 \sin x)dxdy +\\ +24x^2y^2dy^2$