Question

If f has continuous second-order partial derivatives and ∂²f/∂x∂y = g(x, y), find expressions for ∂f/∂x and ∂²f/∂y².

Answer 1

To find ∂f/∂x, differentiate f with respect to x treating y as a constant. To find ∂²f/∂y², differentiate the expression ∂f/∂y with respect to y treating x as a constant.

Step-by-step explanation:

If f has continuous second-order partial derivatives and ∂²f/∂x∂y = g(x, y), we can find expressions for ∂f/∂x and ∂²f/∂y². Let's begin:

1. To find ∂f/∂x, we treat y as a constant and differentiate f with respect to x. This means we differentiate the function f(x, y) with respect to x, treating y as a constant. The result is ∂f/∂x.
2. To find ∂²f/∂y², we treat x as a constant and differentiate ∂f/∂y with respect to y. This means we differentiate the expression ∂f/∂y with respect to y, treating x as a constant. The result is ∂²f/∂y².