Final answer:
The function f(x, y)=(x+y², xy-1) is differentiable at (2,1) because it is composed of polynomials, which are continuous and differentiable everywhere. Its partial derivatives exist and are continuous at the point (2,1), confirming its differentiability there.
Step-by-step explanation:
To show that the function f(x, y)= (x+y², xy-1) is differentiable at the point (2,1), we must verify that the function is continuous and has continuous partial derivatives at that point.
The function is composed of polynomials, which are known to be continuous and differentiable at every point in their domain, implying that f is continuous at (2,1). Furthermore, to verify differentiability, we calculate the partial derivatives of the function and evaluate them at the point (2,1).
Partial derivatives of f are as follows:
- ∂f/∂x = (1, y)
- ∂f/∂y = (2y, x)
At the point (2,1), these partial derivatives are:
- ∂f/∂x (2,1) = (1, 1)
- ∂f/∂y (2,1) = (2, 2)
Since the partial derivatives exist and are continuous at the point (2,1), f is differentiable at that point.