Question

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

Answer 1

$z=ysin(3x+y)$
$z'_x=ycos(3x+y)*3=3ysin(3x+y)$
$z'_y=sin(3x+y)+y*cos(3x+y)*1=sin(3x+y)+ycos(3x+y)$

Answer 2

$z=ycdoysin(3x+y);\ frac{partial}{partial x}(ysin(3x+y))=yfrac{partial}{partial x}(sin(3x+y))=\ =ycos(3x+y)cdotfrac{partial}{partial x}(3x+y)=3ycos(3x+y);\ frac{partial}{partial y}(ysin(3x+y))=sin(3x+y)frac{partial}{partial y}y+yfrac{partial}{partial y}sin(3x+y)=\ =sin(3x+y)cdot1+ycos(3x+y)frac{partial}{partial y}(3x+y)=\ =sin(3x+y)+ycos(3x+y)$