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

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Image taken from Keisler's Elementary Calculus, an infinitesmial approach, third edition, page 56.

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    Yeah .. I would have found this quite confusing if I had been told this when I was learning calculus for the first time ... it will take me several comments to explain the distinction they are trying to make ... simple answer don't worry about it. – 2017-02-25
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    This may depend on what the very words "The differential ..." mean – 2017-02-25

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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)

differentials along two different tangent lines

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: $$dy = f(x, dx)$$