What are the first partial derivatives of the function f(x, y) = x²y?

To find the first partial derivatives of the function f(x, y) = x²y, we need to take the derivative with respect to each variable while treating the other variable as a constant.

1. Partial derivative with respect to x:

We denote this as ∂f/∂x. Here, we differentiate the function with respect to x:

∂f/∂x = ∂(x²y)/∂x

Since y is treated as a constant, we use the power rule:

∂f/∂x = 2xy

2. Partial derivative with respect to y:

We denote this as ∂f/∂y. Now, we differentiate the function with respect to y:

∂f/∂y = ∂(x²y)/∂y

Since x is treated as a constant, the derivative is:

∂f/∂y = x²

In summary:

The first partial derivatives of the function f(x, y) = x²y are:

• ∂f/∂x = 2xy
• ∂f/∂y = x²