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I’ve heard people say $dy/dx$ is not a fraction with $dy$ as the numerator and $dx$ as the denominator; that it is just notation representing the derivative of the function $y$ with respect to the variable $x$. The calculus book I am reading (Calculus: A Complete Course - Adams & Essex) says that even though $dy/dx$, defined as the limit of $\Delta y / \Delta x$, appears to be “meaningless” if we treat it as a fraction; it can still be “useful to be able to define the quantities $dy$ and $dx$ in such a way that their quotient is the derivative $dy/dx$”. It then goes on to define the differential $dy$ as “a function of $x$ and $dx$”, as follows: $$dy = \frac{dy}{dx}\,dx = f’(x)\,dx$$ What is the meaning of $dx$ here? It is now an independent variable, yet it seemingly is not one supplied by most functions I would work with. In a later chapter, focused on using differentials to approximate change, the book gives the following: $$\Delta y = \frac{\Delta y}{\Delta x}\,\Delta x \approx \frac{dy}{dx}\,\Delta x = f’(x)\,\Delta x$$ This makes sense to me. The change in $y$, $\Delta y$, at/near a point can be approximated by multiplying the derivative at that point by some change in $x$, $\Delta x$, at/near the point. Here $\Delta y$ and $\Delta x$ are real numbers, so there is no leap in understanding that is necessary. The problem with the definition of $dy$ is that the multiplication is not between a derivative and a real number, such as in the approximation of $\Delta y$, but a product of a derivative and an object that is not explicitly defined. Because I do not understand what $dx$ is, I can’t use it to build an understanding of what $dy$ is. I also have no real understanding of what $dy$ is meant to be, so I cannot work backwards to attach some meaning to $dx$. Is $dy$ meant to be some infinitesimal quantity? If so, how can we be justified in using it when most of the book is restricted to the real numbers? Are $dy$ and $dx$ still related to the limits of functions, or are they detached from that meaning? Later in the chapter on using differentials to approximate change, the book says it is convenient to “denote the change in $x$ by $dx$ instead of $\Delta x$”. We can just switch out $\Delta x$ for $dx$? Why is it convenient to do this? More importantly, how are we justified in doing this?

What exactly is a differential? And https://math.blogoverflow.com/2014/11/03/more-than-infinitesimal-what-is-dx/ both contain discussions that are beyond my understanding. My problem arose in an introductory textbook, I find it strange that we can just swap out different symbols when we go to great lengths to say they are different entities.

In What is the practical difference between a differential and a derivative? Arturo Magidin’s answer says that it is not “literally true” that $dy = \frac{dy}{dx}\,dx$ and that replacing $\Delta x$ with $dx$ is an “abuse of notation”. If that is the case, then the quotient of $dy$ and $dx$ would not be $\frac{dy}{dx}$, but $\frac{dy}{dx}\frac{dx}{\Delta x}$, right?

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    $\begingroup$ If the goal is only to understand calculus, you can just avoid defining $dx$ and $dy$ and understand calculus very clearly. (This is the approach taken in most real analysis textbooks such as Calculus by Spivak.) Of course, it remains very important to understand that if $\Delta x$ is a tiny (but finite) change the value of the input to a function $f$, and $\Delta y$ is the corresponding change in the value of the output of $f$, then $\Delta y \approx f'(x) \Delta x$. (Later, if we want to develop the subject of Calculus on Manifolds, then we will define something called "differential forms".) $\endgroup$ – littleO Jun 19 at 3:52
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There is a view of calculus that comes from differential geometry, which I find the most intuitive. Here you work with differentials $df$, which are liner maps $$df(x)\delta x \mapsto \delta f$$ where here $x$ is a point in the domain, $\delta x$ is an infinitesimal change in $x$, and $\delta f$ is the corresponding infinitesimal change in $f(x)$. Formally, $\delta x$ is a vector at $x$ in the tangent space of the domain, and $\delta y$ is a vector at $f(x)$ on the tangent space of the codomain.

The differential can be defined in terms of the directional derivative, $$df(x)\delta x = \lim_{t\to 0} \frac{f(x+t\delta x) - f(x)}{t},$$ and the fact that $df(x)$ is linear in $\delta x$ can be proved from the differentiability of $f$.

This view of derivatives as differentials mapping an infinitesimal change to an infinitesimal change is extremely powerful in higher dimensions, but can also apply to the simple case of a real-valued function $y(x):\mathbb{R}\to\mathbb{R}$. We have $$\delta y = dy(x)\delta x,$$ where here $\delta x$ is just a real number (a tangent vector on $\mathbb{R}$ at $x$, or intuitively, an infinitesimal change in $x$) as is $\delta y$, and $dy$ is a $1\times 1$ linear map, i.e. a real number. Writing $y'(x)$ for $dy(x)$ we have that the derivative is literally the quotient $$y'(x) = \frac{\delta y}{\delta x}.$$

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