- #1

- 904

- 1,057

## Homework Statement

Given [itex]f(x,y) = x\cdot 3^{x+y^2} [/itex]. Prove that f is differentiable twice at the point P(1,0).

## Homework Equations

[itex]D\subset\mathbb{R}^2, f\colon D\to\mathbb{R}, P\in \mathring{D}[/itex](interior point) - then f is differentiable n+1 times at P[itex]\Leftrightarrow \exists\varepsilon > 0\colon[/itex] in the sphere [itex]B(P,\varepsilon)[/itex] there are defined n-th degree partial derivatives.

## The Attempt at a Solution

I want to use the definition of a partial derivative [itex]f_x := \lim_{t\to 0}\frac{f(P_x + t, P_y) - f(P_x,P_y)}{t}[/itex] if the limit exists (either in R or +/- infinity).

How can I check if the limit exists for any and all points inside a sphere? I want to give an epsilon and show that every point in the sphere with radius epsilon has a final limit for both f

_{x}and f

_{y}.