Derive Taylor series for df

Question

I was trying to understand ito's lemma. When I came across the Taylor series expansion of $df(x)$.

$df(x) = f'(x) dx + (1/2!) f''(x) (dx)^2 + ...$

I searched everywhere for the derivation of this but couldn't find. So I tried it on my own.

$f(x)= f(a) + f'(a)(x-a) + (1/2!) [f''(a)(x-a)^2] +...$

so if I take $d/dx$

$df(x)/dx = f'(a) + (1/2!) [f''(a)d(x-a)^2/dx] + ...$

Making it

$df(x)/dx = f'(a) + (1/2!) [f''(a) 2(x-a)] + (1/3!) [f''(a) 3(x-a)^2] +...$

Can someone help me with how to proceed from here?

Answer 1

\begin{align} df(x) & = f(x+dx) - f(x). \\ & = \sum\limits_{n=0}^{\infty} \frac {f^{(n)} (x)} {n!} (dx)^n - f(x). \\ & = \sum\limits_{n=1}^{\infty} \frac {f^{(n)} (x)} {n!} (dx)^n.\end{align}