Explanation:
To show that the function f(x,y) = 2x^2 − xy is differentiable at any point (a,b), we need to show that the partial derivatives of f with respect to x and y exist and are continuous at that point.
The partial derivative of f with respect to x is:
∂f/∂x = 4x - y
The partial derivative of f with respect to y is:
∂f/∂y = -x
We can see that both of these partial derivatives are continuous functions of x and y and are defined for all (x,y) in the domain of f. Therefore, f is differentiable at any point (a,b).
The derivative of f at (a,b) is the gradient vector of f evaluated at (a,b), which is given by:
Gradient vector of f(a,b) = ∇f(a,b) = ( ∂f/∂x, ∂f/∂y ) = (4a-b, -a)
So the derivative of f at (a, b) is (4a-b, -a).