Homework Help: How to prove differentiable everywhere?

1. Sep 30, 2011

athrun200

1. The problem statement, all variables and given/known data
See photo, part b and c

2. Relevant equations

3. The attempt at a solution
For part b
It seems it is trival, in part a we have proved that $f_{x}$ and $f_{y}$ exist. Obviously, they are differentiable for x and y$\neq$0

For part c.
It seems there are 2 method to do it.
1. Use first principle.(i.e. take limit)
2.Find $f_{xy}$ and $f_{yx}$
If they equal each other, then f is differentiable at(0,0)

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2. Sep 30, 2011

HallsofIvy

What limit are you talking about? "Differentiability" of a function of two variables is NOT defined by a limit. You might want to look up the definition of "differentiable" for functions of more than one variable.

No, that's not true. There is a theorem that says "If a function is differentiable, on a region, then its mixed second derivatives are equal on that region", but the converse of that statement is not true.

3. Sep 30, 2011

athrun200

How about part b?
You only talk about part c

4. Sep 30, 2011

HallsofIvy

Yes, (b) is trivial- f is a quotient of two polynomials and the denominator is not 0 for any point other than (0, 0).

5. Oct 2, 2011

athrun200

But I saw from wiki that it is.
If not, how do I prove differentiable?

6. Oct 2, 2011

athrun200

I also wonder if $f_{x}$=$\frac{(2^xy^3-6x^2y^2-2x^4)}{(y^4+2x^2y^2+x^4)}$ exist at (0,0).

If not, it seems the question part a has some problems.