Because then, you have to factor it again to make it useful.

Use the product rule, but write it explicitly. Perhaps using the ' character to indicate the derivative for each part. Until you are comfortable, don't try to just emit the entire answer. That's too "formal" (i.e., using a formula to do mathematics) to be useful for learning.

There still may be factoring to do. Likely in fact when trigonometric functions are involved, but it's more than twice the work to distribute the multiplication over the addition before taking the derivative.