The question:

Let f(x,y) be a differentiable function at (0,0) that satisfies f(0,0)=0.
Let \phi(t) be a differentiable function at t=0 that satisfies:
 \phi(0)=0,\phi ' (0)=1 and let g(x,y)=\phi(f(x,y)).

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

My try:
We know the two functions are differentiable at (0,0) , hence:
\phi (\Delta x) = \Delta x + o(| \Delta x | ) and:
 f(\Delta x, \Delta y) =a\Delta x + b\Delta y+ o(||(\Delta x, \Delta y) || )
So:
 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 !