If $f[x]$ and $g[x]$ is any pair of functions that have order of contact $m$ at $x=0$, then what is the order of contact of $f[x^2]$ and $g[x^2]$ at $x=0$?
Mathematica code:
Clear[f, g, x, k, derivativeorder];
m = 3;
Column[Table[{derivativeorder[k], D[f[x^2], {x, k}], D[g[x^2], {x, k}]} /. x -> 0, {k, 1, 2 m}]
This code outputs:
{derivativeorder[1], 0, 0}
{derivativeorder[2], 2f'[0], 2g'[0]}
{derivativeorder[3], 0, 0}
{derivativeorder[4], 12f''[0], 12g''[0]}
{derivativeorder[5], 0, 0}
{derivativeorder[6], 120f^(3)[0], 120g^(3)[0]}
I guessed that the answer was $2m + 1$ which is correct although I don't understand why. I seem to be missing the bigger picture.
Can somebody explain this to me?