Taking the partial derivative of a function and use the Laplace transform on it.

Question

I have the function with 2 variables: f(x,y) = x/y. In order to take the Laplace transform of the function f(x,y) I have to do the following steps:

1. take the derivative of f(x,y) in order to have an Ordinary Differential Equation (ODE).
2. find the ODE from the time-domain to the frequency-domain (or S domain).
3. use the frequency-domain function on the Proportional term.

The derivative of a function with 2 variables is solved by the technic of partial derivative where we treat one variable as a constant and derivate the other. For the first derivative with respect to x we treat z as a constant. And the opposite for the second variable.

$∂f/∂x = x^1/y = 1x^0/y = 1/y$

$∂f/∂y = x * 1/y = x * y^{-1} = x * y^{-1-1} = x * y^{-2} = -x/y^2$

Now that I have the partial derivative of the 2 variables, how do I merge them into one function to take the Laplace transform of it? Actually, I am not sure if I have to merge both partial derivative functions or what else I have to do.