Question

Let f(x, y)=(y², -x²)

Using the definition of differentiability, show that f is differentiable at (1,1)

Answer 1

To show that f(x, y) = (y², -x²) is differentiable at (1, 1), we need to show that the partial derivatives exist and are continuous at (1, 1).

Step-by-step explanation:

To show that f(x, y) = (y², -x²) is differentiable at (1, 1), we need to show that the partial derivatives exist and are continuous at (1, 1). Let's start by calculating the partial derivatives:

∂f/∂x = -2x

∂f/∂y = 2y

Now, substitute (x, y) = (1, 1) into the partial derivatives:

∂f/∂x(1, 1) = -2

∂f/∂y(1, 1) = 2

Since the partial derivatives exist and are continuous at (1, 1), we can conclude that f(x, y) = (y², -x²) is differentiable at (1, 1).