Question

Find all four second partial derivatives for the function f(x,y) = tan⁻¹(y/x).

Answer 1

To find the four second partial derivatives for the function f(x,y) = tan⁻¹(y/x), differentiate the function twice with respect to x and twice with respect to y.

Step-by-step explanation:

To find the four second partial derivatives for the function f(x,y) = tan⁻¹(y/x), we need to differentiate the function twice with respect to x and twice with respect to y. Let's start by finding the first partial derivatives:

∂f/∂x = -y/(x² + y²)

∂f/∂y = x/(x² + y²)

Now, let's find the second partial derivatives:

∂²f/∂x² = (2xy² - 2x³)/(x² + y²)²

∂²f/∂y² = (-2x²y + 2xy³)/(x² + y²)²

∂²f/∂x∂y = (y³ - 3x²y)/(x² + y²)²

∂²f/∂y∂x = (y³ - 3x²y)/(x² + y²)²