The question:

Let f(x,y) be a differentiable function at (0,0) that satisfies f(0,0)=0.

Let $\displaystyle \phi(t)$ be a differentiable function at t=0 that satisfies:

$\displaystyle \phi(0)=0,\phi ' (0)=1 $ and let $\displaystyle g(x,y)=\phi(f(x,y))$.

Prove g is differentiable at (0,0) and that $\displaystyle g_{x}(0,0)=f_{x}(0,0)$.

My try:

We know the two functions are differentiable at (0,0) , hence:

$\displaystyle \phi (\Delta x) = \Delta x + o(| \Delta x | ) $ and:

$\displaystyle f(\Delta x, \Delta y) =a\Delta x + b\Delta y+ o(||(\Delta x, \Delta y) || ) $

So:

$\displaystyle g(\Delta x, \Delta y) = \phi(a\Delta x + b\Delta y+ o(||(\Delta x, \Delta y) || ) ) $

From that point I'm pretty stuck...I can't understand how to continue solving the question...

I'll be delighted to get some guidance

Thanks !