Product rule:
https://en.wikipedia.org/wiki/Product_rule
For three factors:
$$ \frac{d(uvw)}{dx} = \frac{du}{dx}vw + u\frac{dv}{dx}w + uv\frac{dw}{dx} \\ = uvw\left( \frac{1}{u}\frac{du}{dx} + \frac{1}{v} \frac{dv}{dx} + \frac{1}{w}\frac{dw}{dx} \right) $$For $k$ factors, $f_1, f_2, \dots, f_k$ we have:
$$ \frac{d}{dx}\prod_{i=1}^k f_k = \sum_{i=1}^k \frac{df_i}{dx} \prod_{j \ne i} f_j \\ = \left( \prod_{i=1}^k f_i \right) \left( \sum_{i=1}^k \frac{1}{f_i} \frac{df_i}{dx} \right) \\ = \left( \prod_{i=1}^k f_i \right) \left( \sum_{i=1}^k \frac{f'_i(x)} {f_i(x)} \right) $$But we have:
$$ \frac{f'(x)}{f(x)} = f'(\log(x)) $$Therefore:
$$ \frac{d}{dx}\prod_{i=1}^k f_k = \left( \prod_{i=1}^k f_i \right) \left( \sum_{i=1}^k f'_i(\log(x)) \right) $$