Question

Найти полный дифференциал 2 порядка.

Answer 1

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

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

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