Is there a way to distribute a derivative into a product of arbitrary bounds like a summation?

Question

What I am interested in is

$$ \frac{d}{dx}\prod_{a=1}^{b}f_{a}(x). $$ I know that a derivative can easily be distributed into a summation, but what about an arbitrary product?

Answer 1

By using product rule, we have

$$\frac{d}{dx}\prod_{a=1}^b f_a(x) = \sum_{a=1}^b \left(\frac{d}{dx} f_a(x) \right)\prod_{j \ne a}f_j(x)$$