Itô's Lemma: Vanilla Case
State Itô's Lemma (general form) in differential form.
Answer
Let $X_t$ be an Itô process of the form
\[
dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dB_t.
\]
and let $f:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ be twice continuously differentiable in $x$ and once in $t$.
Then Itô's Lemma (differential form) states:
\[
df(X_t,t)
= \Big(f_t + \mu f_x + \tfrac{1}{2}\sigma^2 f_{xx}\Big)\,dt + \sigma f_x\, dB_t.
\]