How can I find all the second order partial derivatives of a given function?

Finding Second Order Partial Derivatives

To find all the second order partial derivatives of a function, we follow a systematic approach. Let’s take a function of two variables as an example:

Suppose we have the function f(x, y) = x^2y + 3xy^2.

Step 1: Calculate First Order Partial Derivatives

First, we need to find the first order partial derivatives with respect to each variable.

• Partial derivative with respect to x:

  fx(x, y) = rac{	ext{d}}{	ext{d}x}(x^2y + 3xy^2) = 2xy + 3y^2

• Partial derivative with respect to y:

  fy(x, y) = rac{	ext{d}}{	ext{d}y}(x^2y + 3xy^2) = x^2 + 6xy

Step 2: Calculate Second Order Partial Derivatives

Now, we will compute the second order partial derivatives by taking the derivatives of the first order partial derivatives.

• Second order partial derivative with respect to x, then x:

  fxx(x, y) = rac{	ext{d}}{	ext{d}x}(fx(x, y)) = rac{	ext{d}}{	ext{d}x}(2xy + 3y^2) = 2y

• Second order partial derivative with respect to x, then y:

  fxy(x, y) = rac{	ext{d}}{	ext{d}y}(fx(x, y)) = rac{	ext{d}}{	ext{d}y}(2xy + 3y^2) = 2x + 6y

• Second order partial derivative with respect to y, then x:

  fyx(x, y) = rac{	ext{d}}{	ext{d}x}(fy(x, y)) = rac{	ext{d}}{	ext{d}x}(x^2 + 6xy) = 2x + 6y

• Second order partial derivative with respect to y, then y:

  fyy(x, y) = rac{	ext{d}}{	ext{d}y}(fy(x, y)) = rac{	ext{d}}{	ext{d}y}(x^2 + 6xy) = 6x

Summary of Second Order Partial Derivatives

For the function f(x, y) = x^2y + 3xy^2, the second order partial derivatives are:

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

By following these steps, you can find the second order partial derivatives for any function of multiple variables.