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In the derivation of the Black-Scholes equation, it is argued (e.g. in the original paper and in Hull) that $$dV(S_t, t)=(…)dt + \frac{\partial V}{\partial S} dS_t,$$ where $V(S_t, t)$ is the value at time $t$ of an option on an asset with price process $S$. Thus, adding an amount of $-\frac{\partial V}{\partial S}$ assets to the option would make the resulting portfolio $P$ riskless since $$dP(S_t, t) \\ =(…)dt + \frac{\partial V}{\partial S} dS_t - d(\frac{\partial V}{\partial S} S_t) \\ =(…)dt + \frac{\partial V}{\partial S} dS_t - \frac{\partial V}{\partial S} dS_t \\ = (…)dt. $$

Why is the second equation true? We are treating $\frac{\partial V}{\partial S}$ like a constant when pulling it out of the derivative while it is actually a function of $t$ and $S$. Wouldn’t we have to use the Ito Product Rule here? (and I wouldn’t know how to do that since I don’t know the dynamics of $\frac{\partial V}{\partial S}$)

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