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The derivative was introduced to clarify the concept of "slope", specifically in the case of curves. It was not created specifically to avoid a "mathematical disaster", but it was possible to formalize the definition of slope (derivative) in such a way that concepts that are not very precise, such as "infinitely small", are avoided. This has been done by defining the derivative in terms of limits.

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8y ago
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8y ago

I am not sure that I understand the question. Perhaps you are referring to division by zero.

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Q: What mathematical disaster does the derivative avoid?
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