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Here's another way to think about this .
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We want to know the speed at exactly t equals 10.
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Speed is delta s, over delta t.
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We would like delta t to be 0,
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so, the only t we're dealing with is t equals 10,
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but we can't have delta t be 0,
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because division by 0
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leads to a quantity that diverges, is not defined,
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just doesn't make sense,
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and so we just sneak up on dividing by 0,
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we let our delta t get
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smaller, and smaller, and smaller, and smaller,
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so we get closer, and closer, and closer
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to dividing by 0,
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this ratio will stay constant in most circumstances
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and we say that that's the instantaneous speed.
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As you probably guessed,
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this instantaneous speed
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or in general, this instantaneous rate of change,
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is known as a derivative.
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I'll define the derivative more carefully in a moment,
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but first I want to give a graphical picture
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to accompany the stories
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and numbers I talked about here.