If you act responsibly, you think about what could go wrong, you keep your promises, and people can trust you with the job. You spend money responsibly when you do not throw it away. You drive responsibly when you are careful with other people’s lives, not only your own. It is the adverb from responsible, and it is a common, useful word — teachers, parents, and news reports all use it. Though actually, it does not mean “able to answer questions” (that is a different, older flavour of responsible); here it means “in a way that shows you can be trusted with a duty.” It is about judgement and care, not about speed.