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)?