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I am supposed to give a 15-20 minutes math lecture, where I am expecting around 20-30 people. The lecture is about derivative. Since this would be my first "class", I would appreciate any suggestions to how to make it interesting.

Students could be bored with definitions, theorems, mathematical concepts that are told in abstract way, therefore my question is:

How to tell a story about derivative in simple, interesting but also, mathematically based way? :)

Any suggestions about mathematical questions, examples, fun problems related to this topic (which should grab their attention) are also welcome.

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Like the good Monsieur Jourdain who didn't know that he spoke prose all his life, you could point out that many in the audience have already seen the derivative many times daily, particularly if they are from well-to-do homes: namely, the speedometer of the family car.

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There is an excellent online courses on coursera entitled Calculus One where the lecturers explain derivatives in many different and interesting ways. Maybe you can use some of that material?

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The derivative of a function is another function that gives a point property of the original function. It is basically about properties of functions and NOT about limits. The role of the limit is limited to implementation. The important aspects of the derivative of a function are what the property is that it gets at, i.e., the steepness of the curve at each value of the independent variable, the basis of our interest in that property, the index of that property that we use - mainly the slope of the tangent line, the formal approach to obtaining it for a given function using a limit, the question of whether that index is even tractable for even just the elementary functions, an indication as given by the slope of a straight line, the curious property of being simple for the elementary functions, the important properties of the derivative that enable us to find derivatives of arithmetic combinations and compositions and implicit forms of the elementary functions very simply, the interesting notations used for the differentiation operator, and the curious relationship of the derivative to the antiderivative and definite integral. Differentiation is a very interesting and useful operation.

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