I know differentials are seen as infinitesimal changes but I am confused because the definition of the differential doesn't seem to imply anything infinitesimal:
If $y = f(x)$ and you are in a point $x$ where the function $f$ has a derivative, and you consider a neighbouring point $x + \Delta x$, then we define the differential $dy$ as the change along the tangent line in $x$, so $dy = f'(x)\Delta x$ .
When taking the identical function $f:y= x$ and apply the definion above, we find $dx = \Delta x$ where $dx$ means the differential of the identical function.
So you can write $dy =f'(x) dx$
But defining differentials like this, they don't have to be small changes at all. They are dependent on the values of $x$ and $\Delta x$.
So far no problem, but then the differential $dx$ also appears when defining integrals as the limit of the Riemann Sums $\Sigma f(x) \Delta x$ where $\Delta x$ goes to 0. We get the well known notation $\int f(x)dx$ where the differential symbol $dx$ magically reappears, but this time it does have the meaning of an arbritry small change of $x$.
So my confusion is, according to the definition , differentials don't have to be infinitesimal, but when they are used in the definition of integrals they suddenly do.
Can anyone clarify this?
