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Do I understand the notion of differential correctly?

Let me put my understanding in my own words:

$ \Delta f(x) = P(x)* \Delta x + Q(\Delta x) $

So, $ P(x)*\Delta x $ is called differential of $f(x)$ function.

Increment of a function $ \Delta f(x) $ can be expressed as a sum of two expressions: function P(x) in x variable and another function $Q(\Delta x)$ in $\Delta x$ variable (we regard $\Delta x$ as a separate variable). So functions P and Q are functions of different variables (and function P is not a function of $\Delta x$).

Besides, $lim (Q(\Delta x))/\Delta x = 0 $ when $\Delta x$ -> 0. Simpler put, $Q(\Delta x)$ is always smaller than $ \Delta x $, given that $\Delta x$ is small.

Is there a flaw in my rewording (internal understanding)?

caasdads
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    Yes, that's pretty much what it is. – zipirovich Jun 12 '18 at 21:08
  • Is the $Q(\Delta x)$ meant to represent the fact that the differentials have parts that we leave off, like in the product rule? If so, then you are correct.

    Another way to think about this is that a differential times a differential or a differential of a differential are both second order infinitesimals - infinitely smaller than an infinitely small value.

     – johnnyb Jun 13 '18 at 01:33

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