Itô's lemma

Taylor expand. For a smooth $F$, the ordinary second-order Taylor expansion in both variables over a small step $\Delta t$ with $\Delta W=W_{t+\Delta t}-W_t$ reads $\Delta F\approx \partial_tF\,\Delta t+\partial_xF\,\Delta W+\tfrac12\partial_x^2F\,(\Delta W)^2+\tfrac12\partial_t^2F\,(\Delta t)^2+\partial_t\partial_xF\,\Delta t\,\Delta W+\cdots$ — this much is pure calculus, valid for any smooth path.

Order the terms by size. Because $\Delta W$ is of order $\sqrt{\Delta t}$ (not $\Delta t$, as shown in the previous theorem), the terms scale as: $\partial_tF\,\Delta t$ is order $\Delta t$; $\partial_xF\,\Delta W$ is order $\sqrt{\Delta t}$ (the dominant, leading-order random term); $(\Delta t)^2$ and $\Delta t\,\Delta W$ are order $(\Delta t)^2$ and $(\Delta t)^{3/2}$ respectively, negligible compared to $\Delta t$; but $(\Delta W)^2$ is order $\Delta t$ — the same order as $\partial_tF\,\Delta t$, not negligible at all, unlike in ordinary calculus where $(\Delta x)^2$ is always negligible next to $\Delta x$.

Replace $(\Delta W)^2$ by its mean. By the quadratic variation theorem, summed over many small steps $(\Delta W)^2$ behaves like $\Delta t$ (its mean, with fluctuations around that mean vanishing as the steps shrink and are summed), which is the informal justification for the heuristic substitution rule $(dW_t)^2=dt$ in the limit of infinitesimal steps.

Take the limit. Dropping the negligible higher-order terms and substituting $(\Delta W)^2\to dt$ in the limit $\Delta t\to0$ turns the Taylor expansion into the differential form $dF=\Big(\partial_tF+\tfrac12\partial_x^2F\Big)dt+\partial_xF\,dW_t$ exactly as claimed — the $\tfrac12\partial_x^2F\,dt$ term is precisely the contribution that ordinary calculus discards but stochastic calculus must keep.