continuity and differentiability

Hello I would just like you to tell me what would be the best method to demonstrate that the function g is of class on the interval J(I was spoken to apply Taylor Lagrange or Taylor-Young but I did not understand much) I thank:

Here is the statement:

Let f be a function of class over an interval J, such that f (0) = 0.

It is assumed that 0 is an interior point to J. We define the function g on J by:

if

and g (0) = f '(0).

What I have done yet below:

In a first time I apply Taylor Young to see where that leads me

-The formula for Taylor Young in 0 to order 2 of f gives us

or f (0) = 0 therefore

and

I wonder now if this is fair and whether it will indeed serve me something.

(I just want to say that I'm French and I think you'll find my language can be a little too basic, I hope that this will not prevent you from answering me)