Question

Does anyone have a simple counter example showing why dy/dx cannot be treated as a ratio?

Answer 1

It turns out that in general it CAN be treated as a ratio. We see this when we learn the chain rule, and it becomes more apparent when we dip into the topic of differential equations. Yet we're told in no uncertain terms that it is not a ratio, and this point is stressed so that we understand the definition of dy/dx as a limit and not as the result of dividing one quantity by another.

Answer 2

Do you have any example of math breaking down because of misapplication of dy/dx as a ratio?

Answer 3

I want to introduce the derivative of ln(x) along with implicit differentiation and would like a sample "cautionary tale" of sorts.