After a producer determines that the demand for one of its products is inelastic, why would this firm probably raise the price of this product?

The reason why this firm probably raise the price of this product is: The firm's total revenue would increase.

Inelastic demand indicated that the demand of that specific products will not be affected by the price fluctuation, mainly because it is either rare or a very important commodity.

Since the amount of the demand wouldn't be affected by the price, the firm could go all the way in setting up their price.