Question

Let f(x, y)=(xy,2x+y²)

Using the definition of differentiability, show that f is differentiable at every point of R².

Answer 1

To show that the function f(x, y) is differentiable at every point in ℝ², we analyze the continuity of its partial derivatives, which are all polynomials and thus continuous everywhere in ℝ², confirming the function's differentiability.

Step-by-step explanation:

The question is asking to show that the function f(x, y) = (xy, 2x + y²) is differentiable at every point in ℝ² (the set of all points on the plane with real number coordinates). To demonstrate this, we need to show that the partial derivatives of f exist and are continuous throughout the domain.

The function f has two components, f₁(x, y) = xy and f₂(x, y) = 2x + y². For differentiability, we need to check the partial derivatives of these components with respect to x and y. The partial derivative of f₁ with respect to x is y, and with respect to y is x. Similarly, the partial derivatives of f₂ with respect to x is 2, and with respect to y is 2y.

Since all these partial derivatives are polynomials in x and y, they are continuous everywhere in ℝ². This means that f is indeed differentiable at every point in ℝ².