Compute all the first and second partial derivatives of the function. We assume that y>0𝑦>0 .f(x,y)=xy−3x−ln(y6) .

First, let's compute the first partial derivatives:

1. Partial derivative with respect to x (∂f/∂x):

   ∂f/∂x = y - 3

2. Partial derivative with respect to y (∂f/∂y):

   ∂f/∂y = x - 6/y

Now, let's compute the second partial derivatives:

1. Second partial derivative with respect to x (∂²f/∂x²):

   ∂²f/∂x² = 0

2. Second partial derivative with respect to y (∂²f/∂y²):

   ∂²f/∂y² = 6/y²

3. Mixed second partial derivative with respect to x and y (∂²f/∂x∂y):

   ∂²f/∂x∂y = 1

4. Mixed second partial derivative with respect to y and x (∂²f/∂y∂x):

   ∂²f/∂y∂x = 1

So, the first and second partial derivatives of the function f(x,y)=xy−3x−ln(y6) are as above.