To prefix all tailwind classes, you can add the prefix: 'tw-' option in your tailwind.config.js: https://tailwindcss.com/docs/configuration#prefix
If you already have Mix configured to handle tailwind, there's nothing else you need to do.
If you're using Tailwind 3, it uses the JIT compiler by default so there's no need to purge your CSS. You just have to make sure your conent array is set correctly in your config: https://tailwindcss.com/docs/upgrade-guide#configure-content-sources. Also, make sure to have npm run watch running so it automatically generates classes as you write them.
If you're using Tailwind 2, make sure you have the purge array set instead: https://v2.tailwindcss.com/docs/optimizing-for-production#writing-purgeable-html.
If you want to purge styles in the other CSS framework you're using, you're going to have to do it manually with purgecss. This package is popular in getting it work nicely with mix: https://github.com/spatie/laravel-mix-purgecss.