For any specific derivative, you can ask a computer to check your result, as several other answers suggest.
However, if you want to be self-sufficient in taking derivatives (for an exam or other work), I recommend lots of focused practice. Most calculus textbooks include answers to the odd-numbered problems in the back of the book, and if you search for "derivative worksheet" you'll find lots of problem lists online.
Work through a list of at least 20 problems, and check your answers-- if you get less than 80% or 90% right, you know you need more practice. Here's the most important part: Track down your mistakes. Watch out for them in the future, and be sure you understand the right way to go. Pay attention to simplifying your answers, too, because a lot of people make algebra mistakes after getting the calculus right.
The rules you have are the best way to take these derivatives, you just have to be able to use them accurately.