What is meant by the last paragraph? What is meant by the last paragraph you see further below? "The differential..."

Image taken from Keisler's Elementary Calculus, an infinitesmial approach, third edition, page 56.
 A: As Hagen von Eitzen says, it depends on exactly how Keisler is defining differentials. There have been numerous different schemes produced to give a solid definition to the differential. Since I don't have Keisler's book, I do not know which he is using. But based on the few clues here, I am guessing that his differentials are coordinates along the tangent line (or at least equivalent to them): (pardon the crudity of the Paint-produced graph)

Start with the graph of $y = f(x)$ and look at some point $x_1$. The tangent line is the line passing through $(x_1, f(x_1))$ with slope $f'(x_1)$. Then we think of $dx$ and $dy$ as being the $x$ and $y$ coordinates along this line, except that the origin of the $(dx, dy)$ coordinates is at the point of tangency. So each pair of values $(dx, dy)$ specifies a point on this tangent line. If I know the value $dx$ of some point on the line, I can get $dy$ from the point-slope equation for that line (remembering that in the differential coordinate system, the point of tangency is $(0,0)$): $$dy - 0 = f'(x_1)(dx - 0)$$
which clearly simplifies to $f'(x) = \frac{dy}{dx}$ when $dx \ne 0$.
But what happens if we take a different value for $x$, say $x_2$? We get a different tangent line, with a different slope. So even if we choose the same value for $dx$, we get a different value for $dy$:
$$dy - 0 = f'(x_2)(dx - 0)$$
In the graph for the same differential value $dx$, we see that the differential value $dy_1$ of the tangent line at $x = x_1$ is positive. But the differential value $dy_2$ of the tangent line at $x = x_2$ is both negative and larger in magnitude than $dy_1$.
Thus the differential $dy$ depends not only on $dx$ but also on the location $x$ where the tangent line is taken. It can be considered a function of both values:
$$dy = f(x, dx)$$
