(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f(x,y)=x^{3}+ x^{2}y + y^{3}. Show that f is differentiable at (1,2).

2. Relevant equations

A function f(x,y) is differentiable at a point (x_{0},y_{0}) if the partial derivatives of f(x,y) are defined at (x_{0},y_{0}) and if the following limiting condition holds:

lim_{(x,y)-->(x0,y0)}[ f(x,y) - f(x_{0},y_{0}) - ∂f/∂x|_{(x0,y0)}(x-x_{0}) - ∂f/∂y|_{(x0,y0)}(y-y_{0}) ] / √( (x-x_{0})^{2}+ (y-y_{0})^{2}) = 0.

3. The attempt at a solution

∂f/∂x = 3x^{2}+ 2xy ===> ∂f/∂x|_{(1,2)}defined

∂f/∂y = 2x^{2}+ 3y^{2}===> ∂f/∂y|_{(1,2)}defined

But I'm having trouble with making lim_{(x,y)-->(1,2)}= 0.

I got it all the way to

lim_{(x,y)-->(1,2)}= (x^{3}+x^{2}y+y^{3}+22 -7x-13y)/√((x-1)^{2}+(x-2)^{2}).

I need to get rid of the denominator. I thought about rationalizing the denominator, but I don't that'll get me anywhere.

Suggestions welcome.

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# Homework Help: Is this f(x,y) differential at (1,2)?

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