Find all the second order partial derivatives of the following function

To find the second order partial derivatives of a function, we first need to determine the first order partial derivatives with respect to each variable. Let’s consider a function f(x, y).

1. **First Order Partial Derivatives:** Start by calculating the first order partial derivatives:

• Partial derivative with respect to x: f_x = ∂f/∂x
• Partial derivative with respect to y: f_y = ∂f/∂y

2. **Second Order Partial Derivatives:** Once you have the first order derivatives, compute the second order derivatives:

• Second partial derivative with respect to x: f_xx = ∂²f/∂x²
• Second partial derivative with respect to y: f_yy = ∂²f/∂y²
• Mixed partial derivative with respect to x and y: f_xy = ∂²f/∂x∂y
• Mixed partial derivative with respect to y and x: f_yx = ∂²f/∂y∂x

3. **Example:** As an example, let’s take the function f(x, y) = x²y + 3xy².

• First, calculate f_x and f_y:
• f_x = 2xy + 3y²
• f_y = x² + 6xy

4. **Now calculate the second order partial derivatives:**

• f_xx = ∂/∂x (2xy + 3y²) = 2y
• f_yy = ∂/∂y (x² + 6xy) = 6x
• f_xy = ∂/∂y (2xy + 3y²) = 2x + 6y
• f_yx = ∂/∂x (x² + 6xy) = 6y

In summary, for the function f(x, y) = x²y + 3xy², the second order partial derivatives are:

• f_xx = 2y
• f_yy = 6x
• f_xy = 2x + 6y
• f_yx = 6y