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I was reading Leithold and the introduction to differentials deals with approximations, $\Delta y \approx f’(x) \Delta x$. This approximation is usable for small values of $\Delta x$.

Then the definition is made using a equality: $dy = f’(x)dx$.

Please explain this jump from $\approx$ to $=$

cgo
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  • One is $\Delta y$ and the other is $dy$. There's no conflict. – saulspatz Mar 11 '20 at 00:01
  • $dx$ is used when you consider the limit of $\Delta x \to 0$. In this process, the error appearing in the approximation, vanishes. Thus, with no error (the error is zero),, approximation is actually an equality- – Crostul Mar 11 '20 at 00:01

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What do you even mean by $dy=f'(x)dx$? Can you explain what is the definition of $dx,dy$ and $f'(x)$?

Jingeon An-Lacroix
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