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taiyo comments on Open Thread June 2010, Part 4 - Less Wrong
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I'm not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly''.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan "brought to you by apostrophe, the letters LIM and an arrow" but you can go on to tell them "this is how impressive mathematcians say you should write this stuff that you already understand" and they'll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn't easy.
That is close to saying "this stuff is hard". How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?