Частная производная функции z=y*sin(3x+y) ?

[latex]z=ysin(3x+y)[/latex] [latex]z'_x=ycos(3x+y)*3=3ysin(3x+y)[/latex] [latex]z'_y=sin(3x+y)+y*cos(3x+y)*1=sin(3x+y)+ycos(3x+y)[/latex]

[latex]z=y\cdoy\sin(3x+y);\\ \frac{\partial}{\partial x}(y\sin(3x+y))=y\frac{\partial}{\partial x}(\sin(3x+y))=\\ =y\cos(3x+y)\cdot\frac{\partial}{\partial x}(3x+y)=3y\cos(3x+y);\\ \frac{\partial}{\partial y}(y\sin(3x+y))=\sin(3x+y)\frac{\partial}{\partial y}y+y\frac{\partial}{\partial y}\sin(3x+y)=\\ =\sin(3x+y)\cdot1+y\cos(3x+y)\frac{\partial}{\partial y}(3x+y)=\\ =\sin(3x+y)+y\cos(3x+y)[/latex]