Question

Найти частные производные второго порядка функции z=arcsin(x-y)

Answer 1

$z=arcsin(x-y)\\\\z'_{x}=\frac{1}{\sqrt{1-(x-y)^2}}\\\\z''_{xx}=-\frac{1}{2}\cdot (1-(x-y)^2)^{-\frac{3}{2}}\cdot (-2(x-y))=\frac{2(x-y)}{2\sqrt{(1-(x-y)^2)^3}}\\\\z''_{xy}=-\frac{1}{2\cdot \sqrt{(1-(x-y)^2)^3}}\cdot (-2(x-y))\cdot (-1)=-\frac{x-y}{\sqrt{(1-(x-y)^2)^3}}\\\\z'_{y}=-\frac{1}{\sqrt{1-(x-y)^2}}\\\\z''_{yy}=\frac{x-y}{\sqrt{(1-(x-y)^2)^3}}$