Verify Euler’s theorem for the function f(x,y)=30(x^0.5)(y^1.5)-2(x^3)(y^(-1))

To verify Euler's theorem for a function f(x, y), we need to calculate the partial derivatives of the function with respect to x and y and then check if they satisfy the equation:

df/dx = (∂f/∂x)dx + (∂f/∂y)dy
df/dy = (∂f/∂x)dx + (∂f/∂y)dy

where dx and dy are infinitesimally small changes in x and y respectively.

Let's calculate the partial derivatives of the function f(x, y):

∂f/∂x = (30 * 0.5 * x^(-0.5) * y^1.5) - (2 * 3 * x^2 * y^(-1))
= 15 * x^(-0.5) * y^1.5 - 6 * x^2 * y^(-1)

∂f/∂y = (30 * x^0.5 * 1.5 * y^0.5) - (2 * x^3 * (-1) * y^(-2))
= 45 * x^0.5 * y^0.5 - (-2) * x^3 * y^(-2)
= 45 * x^0.5 * y^0.5 + 2 * x^3 * y^(-2)

Now, let's substitute these partial derivatives back into the equations:

df/dx = (15 * x^(-0.5) * y^1.5 - 6 * x^2 * y^(-1)) * dx + (45 * x^0.5 * y^0.5 + 2 * x^3 * y^(-2)) * dy

df/dy = (15 * x^(-0.5) * y^1.5 - 6 * x^2 * y^(-1)) * dx + (45 * x^0.5 * y^0.5 + 2 * x^3 * y^(-2)) * dy

Notice that the terms with dx and dy are the same in both equations, which satisfies Euler's theorem. Therefore, Euler's theorem is verified for the function f(x, y) = 30(x^0.5)(y^1.5) - 2(x^3)(y^(-1)).