Finding the Second Partial Derivative of a Multivariable Function

[chemphys1]

Homework Statement

Show z(x,y) = cos(xy) is a solution of

(∂z/∂x)_y + (∂z/dy)_x = (x+y) ( (∂²z/∂x∂y) + xyz)

(question also attached if it makes it clearer)

The Attempt at a Solution

∂z= (∂z/∂x)_ydx + (∂z/dy)_xdy

∂z/∂x = -ysin(xy)
∂z/∂y = -xsin(xy)

what does it mean show it is a solution? any tips appreciated

 

[CAF123]

what does it mean show it is a solution? any tips appreciated

It means that choice of z(x,y) satisfies the equation. I.e plug in z(x,y) into LHS and into the RHS and they should be equal.

 

[Joffan]

So for $z(x,y) = \cos(xy)$, what is $\frac{∂^2z}{∂x∂y}$?