When we specify a GBM stock price:
$$dS = \mu S dt + \sigma S dW$$
And then we change it to: $$\frac{dS}{S} = \mu dt + \sigma dW$$
The we assume: let $Z_t = f(S_t) = \log S_t$, where $f(x) = \log x$. Then by the Ito's formula, we have:
$$ Z_t - Z_0 = f(S_t) - f(S_0) = \int_{0}^{t} f'(S_s) \; dS_s + \int_{0}^{t} \frac{1}{2} f''(S_s) \; dS_s^2,$$
What happened to $\frac{\partial Z_t}{\partial t} $ in Ito ? Ito gives: $$ dZ_t = \frac{\partial Z_t}{\partial t}dt + \frac{\partial Z_t}{\partial S}dS + \frac{1}{2}\frac{\partial^2 Z_t}{\partial S^2}dS^2$$
and somehow we assumed that this quantity $\frac{\partial Z_t}{\partial t} $ is null.
