Assumption in black scholes solution

Question

Under the usual notations,

In most textbooks on Quantative Finance, for deriving the Black-Scholes solution I find that authors, while setting up the riskless portfolio, assume that,

$$\text{d} (\frac{\partial V}{\partial S} S_t) = \frac{\partial V}{\partial S} \text{d} S_t $$

At least can we prove this post facto, as in, does this equation hold true for famous Black Scholes equation

The same issue is also pointed out here.

Answer 1

This is not true. Note that $\frac{\partial C}{\partial S_t} = N(d_1)$. Then \begin{align*} d\left(\frac{\partial C}{\partial S_t}S_t\right) &= \underbrace{S_t dN(d_1) + d\langle N(d_1), S\rangle_t} + N(d_1) dS_t\\ &\ne N(d_1) S_t. \end{align*} That is, \begin{align*} d\left(\frac{\partial C}{\partial S_t}S_t\right)\ne \frac{\partial C}{\partial S_t}dS_t. \end{align*}