Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 25

$z_x=8xy^3e^{x^2y^3}$ $z_y=12x^2y^2e^{x^2y^3}$

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: $z_x=4e^{x^2y^3}*2xy^3=8xy^3e^{x^2y^3}$ $z_y=4e^{x^2y^3}*3x^2y^2=12x^2y^2e^{x^2y^3}$