How do we verify the Schwarz theorem on f(x;y) = (2x+3y) / (cos(y) +sin(x))?

 The Schwarz theorem states that for a continuously differentiable function f(x,y), the mixed partial derivatives fxy and fyx are equal. To verify the Schwarz theorem on the function f(x,y) = (2x+3y) / (cos(y) + sin(x)), we can compute the partial derivatives and check if they are equal.

Let's first compute the partial derivatives of f(x,y):

fx = 2/(cos(y) + sin(x))

fy = 3/(cos(y) + sin(x)) * (-sin(y))

To compute the second-order partial derivatives, we need to differentiate fx with respect to y and fy with respect to x:

fxy = -2 sin(y)/(cos(y) + sin(x))^2

fyx = -2 sin(y)/(cos(y) + sin(x))^2

We can see that fxy and fyx are equal. Therefore, the Schwarz theorem holds for the function f(x,y) = (2x+3y) / (cos(y) + sin(x)).