Generalizing the Product Rule

Question

How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for

$$ \frac{d}{dx} \prod_{j = 1}^n \psi_j(x) $$

Answer 1

You have

$$\frac{d}{dx} \prod_{j=1}^n \psi_j(x) = \sum_{k=1}^n \left(\psi_k'(x)\prod_{\substack{j=1\\j \neq k}}^n \psi_j(x)\right).$$

If none of the $\psi_j$ has zeros, you can also write it in the form

$$\frac{d}{dx} \prod_{j=1}^n \psi_j(x) = \left(\sum_{k=1}^n \frac{\psi_k'(x)}{\psi_k(x)}\right)\prod_{j=1}^n \psi_j(x).$$

Answer 2

$(fgh)'=f'gh+f(gh)'=f'gh+fgh'+fg'h$. Already we can see how this will go. Can you prove it by induction?