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> To visualize derivatives, we can draw a right triangle whose hypoteneuse [sic] is tangent to a function. If the triangle's width is , then its height is the derivative." is the most confusing and unintuitive way to visualize derivatives I can think of

This isn't only unintuitive, it's literally wrong.

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No. When you first start learning about derivatives, you are told you draw a line tangent to a point and then find the slope of that line at that point. The slope is just del y / del x. So, if del x = 1, then the slope is just del y / 1 = del y = the height of the triangle.

Of course, it is more useful to get a derivative as a function of x, so you don't have to do this at every point. But, as a starting point, when you are first learning about derivatives and only have a background in algebra and geometry, this is a nice way to think about it.

Right. My browser didn't play nice with MathML, so I missed about the part where x = 1 (i.e. I couldn't see the "1" :) )

My objection was that a tangent line, as opposed to a segment, has infinite length. I can draw infinitely many segments tangent to the curve -- and get infinitely many heights for it.

Ah, ok. That makes sense. Confusion resolved!

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