What is the derivative of an iterated product like $ \frac{d}{dx}\prod \limits_{i=0}^n \ln(y_i^{x - 1})$? Suppose the following function with pi notation, with the pi denoting the iterated product, multiplying from $i = 0$ to $i = n$: $$\prod_{i=0}^n \ln(y_i^{x - 1})$$ That is, the natural logarithm of $y$, subscripted by $i$, to the power of $x - 1$. What is the derivative of this product - to be clear, its derivative with respect to $x$, not $y$?

We don't even need a product rule. We have \begin{align*} g(x) = \prod_{i=0}^{n}\ln(y_i^{x-1}) = (x-1)^{n+1}\prod_{i=1}^{n}\ln(y_i) \end{align*} And so \begin{align*} \frac{d}{dx} g(x) = (n+1)(x-1)^{n}\prod_{i=0}^{n}\ln(y_i) \end{align*}