Question

u=x sin xy+y cos xy. Найти d^2/dx^2

Answer 1

$u=x\cdot sinxy+y\cdot cosxy\\\\\frac{\partial u}{\partial x}=sinxy+x\cdot cosxy\cdot y+y\cdot (-sinxy)\cdot y=\\\\=sinxy+xy\cdot cosxy-y^2\cdot sinxy=(1-y^2)\cdot sinxy+xy\cdot cosxy\\\\\frac{\partial ^2u}{\partial x^2}=(1-y^2)\cdot cosxy\cdot y+y\cdot cosxy+xy\cdot (-sinxy)\cdot y=\\\\=(1-y^2)\cdot y\cdot cosxy+y\cdot cosxy-xy^2\cdot sinxy=\\\\=(2-y^2)\cdot y\cdot cosxy-xy^2\cdot sinxy$