Interchanging calculus operations with pi product In the domain of convergence, we can interchange derivatives, sums and integrals but what about $ \prod$ and previous operations?
For example,
$$ \frac{d}{dx} \sum_{j=1}^{n} f_j(x) =  \sum_{j=1}^{n} \frac{d}{dx}f_j(x)$$
and,
$$ \frac{d}{d u^i} \int_{a}^{b} F(u_1,u_2,u_3..) du_j = \int_{a}^{b} \frac{d}{du^i}  F(u_1,u_2,u_3..) du_j $$
for $ i \neq j$
But, how would I interchange product and derivative like:
$$ \frac{d}{dx} \prod_{i=1}^{i=n} f_i(x)=?$$
One indirect way I did was this:
$$ g(x) = \prod_{i=1}^{i=n} f_i(x)$$
Take log of both sides and then,
$$ g'(x) = g(x) \sum_{i=1}^{i=n} \frac{ f_i^{'}(x) }{ f_i (x)}$$
Would there be more direct interchanges as I had shown before/ alternate proofs of this identity?

The formula for the derivative of a product gives you
$$\frac{d}{dx} \prod_{j=1}^{n} f_i(x)=  \sum_{j=1}^n f'_j(x)\prod_{i=1 \\ i\neq j}^{n} f_i(x).$$
