What is the third derivative of $\prod _{i=0}^n\:\left(x-x_i\right)$?

Question

$$\frac{d^3}{dx^3}\left(\prod \:_{i=0}^n\:\left(x-x_i\right)\right)$$

How can I calculate this? $x_i$ are numbers, not variables, the only variable here is x

Answer 1

$$\left(\prod_{i=1}^n(x-x_i)\right)'=\sum_{j=1}^n\prod_{i=1,\\i\ne j}^n(x-x_i)$$

$$\left(\prod_{i=1}^n(x-x_i)\right)''=\sum_{j=1}^n\sum_{k=1}^n\prod_{i=1,\\i\ne j\\i\ne k}^n(x-x_i)$$

$$\left(\prod_{i=1}^n(x-x_i)\right)'''=\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n\prod_{i=1,\\i\ne j\\i\ne k\\i\ne l}^n(x-x_i)$$

Answer 2

Hint: Take $$y=\left(\prod \:_{i=0}^n\:\left(x-x_i\right)\right)$$ then $$\ln y=\sum_{i=0}^n \ln(x-x_i)$$ and $$\frac{y'}{y}=\sum_{i=0}^n \frac{1}{x-x_i}$$ this makes it easy for another derivative.