Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 28

$z_x=ye^{xy}sin(4y^2)$ $z_y=8ye^{xy}cos(4y^2)+xe^{xy}sin(4y^2)$

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: $z_x=e^{xy}sin(4y^2)*y=ye^{xy}sin(4y^2)$ $z_y=e^{xy}*cos(4y^2)*8y+e^{xy}sin(4y^2)*x=8ye^{xy}cos(4y^2)+xe^{xy}sin(4y^2)$