From pg. 74 of No-Nonsense Quantum Mechanics, the author derives the canonical commutation relation from the momentum operator $\hat{p}_i$ as follows:
Question: How does the product rule (for complex derivatives, which I assume is what the author is referring to) imply that
$$ (-i \hslash \partial_i \hat{x}_j + \hat{x}_j i \hslash ) | \Psi\rangle $$
equals
$$ -(i \hslash \partial_i \hat{x}_j) | \Psi \rangle - \hat{x}_j (i \hslash \partial_i | \Psi \rangle) + \hat{x}_j i \hslash \partial_i | \Psi \rangle $$
$$ -i \hslash \partial_i \hat{x}_j |\Psi\rangle = -i \hslash \partial_i (\hat{x}_j |\Psi\rangle) = -i \hslash \partial_i (x_j |\Psi\rangle) \\ = \{ \text{ product rule on factors $x_j$ and $|\Psi\rangle$ } \} \\ = -i \hslash ((\partial_i x_j) |\Psi\rangle) + x_j (\partial_i |\Psi\rangle) $$