What is the motivation behind this definition?

Question

What is the motivation behind this definition? Could you give me an intuititive explanation?

We may also apply the differential notation to terms. If $\tau(x)$ is a term with the variable $x$, then $\tau(x)$ determines a function $f$.

$$\tau(x)=f(x),$$ and the differential $d\tau(x)$ has the meaning $$d\tau(x)=f'(x)dx\;.$$

Look at the bottom of page 58 and top of page 59 (and not 100-101) of https://www.math.wisc.edu/formMail/throttle.php?URL=/~keisler/chapter_2a.pdf

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EDIT

As @Erick Wong so nicely adds in a comment: "The definition simply extends the meaning of $dy$ (where $y$ is a function of $x$) to an arbitrary expression that depends on $x$. ... But as in Example 5(a) it is natural to want to write $d(x^3) = 3x^2\, dx$ directly without first having to define a new function $y = x^3$ and then writing that as $dy$."

Answer 1

A somewhat shorter explanation: your equation says the same thing as $f'(x)=\frac{dy}{dx}$ or $dy=f'(x)dx$.

Answer 2

Let $y= x^2\rightarrow (1)$

We assume that if $x$ is incremented by $dx$ then $y$ changes by $dy$ . If that's true then we have,

$y + dy = ( x+ dx)^2 \rightarrow (2).$

$$(2)-(1) = dy = 2xdx + (dx)^2$$

now $dx\rightarrow 0$ so $(dx)^2$ is even smaller hence $dy =2xdx$.

Similarly we go on for other functions. But its not possible to use this method to calculate differentials because when the functions are not polynomials or are composite then the method becomes difficult.

Just as we can use sum of series to calculate integrals. But its not possible to calculate the sum of every series that why we use the fundamental theorem of calculus. If you were asking this I hope it helps!