Answer
$g_x=-2ysin(2xy)$
$g_y=-2xsin(2xy)$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat y as a constant, and vice versa:
$g_x=-sin(2xy)\times2y=-2ysin(2xy)$
$g_y=-sin(2xy)\times2x=-2xsin(2xy)$