The second-order total differential of a function of two variables is defined and obtained from the first order differential. If z = f (x,y), the first-order total differential of z is

dz = f_x dx + f_y dy.

The second-order total differential of z, denoted by d²z, is given by d²z = d (dz). Thus

d²z =d (f_x dx + f_y dy)

= ∂/∂x (f_x dx + f_y dy) dx + ∂/∂y (f_xdx + f_y dy) dy

= [f_xx dx + f_x ∂/∂x (dx) + f_yx dy + f_y ∂/∂x (dy)] dx

+ [f_xy dx + f_x ∂/∂y (dx) + f_yy dy + f_y ∂/∂y (dy)] dy

However, dx and dy are considered as constants, so

∂/∂x (dx) = 0, ∂/∂x (dy) = 0, ∂/∂y (dx) = 0, ∂/∂y (dy) = 0

d²z = (f_xxdx + f_xy dy) dx + (f_xy dx + f_yy dy) dy
                                          (assuming f_xy = f_yx)

= f_xx (dx)² + f_xy dx dy + f_xy dx dy + f_yy (dy)²

= f_xx (dx)² + 2f_xy dx dy + f_yy (dy)² .

Hence the second-order total differential of z is

d² z = f_xx (dx)² + 2f_xy dx dy + f_yy (dy)².

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