Question

For the function f(x, y) = √(3x2 + 4y2), what is the partial derivative ∂f/∂x2 at (x, y)?

Answer 1

To find the partial derivative ∂f/∂x2 at (x, y) for the given function f(x, y) = √(3x2 + 4y2), differentiate the function with respect to x twice while treating y as a constant. The final result is ∂²f/∂x2 = 3 / √(3x2 + 4y2) - (9x2 / (3x2 + 4y2)3/2).

Step-by-step explanation:

To find the partial derivative ∂f/∂x2 at (x, y) for the given function f(x, y) = √(3x2 + 4y2), we need to differentiate the function with respect to x twice while treating y as a constant.

Starting with the first derivative, we get ∂f/∂x = (1/2) * (3x2 + 4y2)-1/2 * 6x = 3x / √(3x2 + 4y2).

Now, differentiating again, we find ∂²f/∂x2 = (d/dx) (3x / √(3x2 + 4y2)) = 3 / √(3x2 + 4y2) - (9x2 / (3x2 + 4y2)3/2).