Explanation of Derivatives?

I tend to be a fairly logical, engineering and scientific person, so I like things concrete and factual. I find abstract concepts hard to grasp. So calculus is definitely a challenge for me. Can anyone offer a clear, concise, helpful explanation of what a derivative of a function is? What does the new function represent, and what is its relation to the original? If you can provide a visual beyond graphs of rates of change and tangent slope graphs, that would be great too! If at all possible, can you explain it with as few mentions of limits, tangent graphs, and rate of change problems as possible? I need something I can visualize. Well, that is unless you can give me a picture using those things that doesn't involve that same graph I keep seeing but not understanding. Thanks in advance for your time and help!