# Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $$\frac{\textrm{d}y}{\textrm{d}x}$$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $$\textrm{d}y = f'(x)\textrm{d}x$$ we are able to plug in values for $$\textrm{d}x$$ and calculate a $$\textrm{d}y$$ (differential). Then if we rearrange we get $$\frac{\textrm{d}y}{\textrm{d}x}$$ which could be seen as a ratio.

I wonder if the author say this because $$\mbox{d}x$$ is an independent variable, and $$\textrm{d}y$$ is a dependent variable, for $$\frac{\textrm{d}y}{\textrm{d}x}$$ to be a ratio both variables need to be independent.. maybe?

• It may be of interest to record Russell's views of the matter: "Leibniz's belief that the Calculus had philosophical importance is now known to be erroneous: there are no infinitesimals in it, and $dx$ and $dy$ are not numerator and denominator of a fraction." (Bertrand Russell, RECENT WORK ON THE PHILOSOPHY OF LEIBNIZ. Mind, 1903). Commented May 1, 2014 at 17:10
• math.stackexchange.com/questions/1548487/…
– user117644
Commented Dec 26, 2015 at 22:58
• So yet another error made by Russell, is what you're saying? Commented Apr 29, 2017 at 4:24
• It can be roughly interpreted as a rate of change of $y$ as a function of $x$ . This statement though has many lacks. Commented Aug 28, 2017 at 17:39
• Obviously it is a ratio! You can tell by the fact that $$\frac{dy}{dx}$$ is obtained with \frac{dy}{dx}, and frac is for "fraction". (I am joking.) Commented Oct 10, 2022 at 4:04

Historically, when Leibniz conceived of the notation, $\frac{dy}{dx}$ was supposed to be a quotient: it was the quotient of the "infinitesimal change in $y$ produced by the change in $x$" divided by the "infinitesimal change in $x$".

However, the formulation of calculus with infinitesimals in the usual setting of the real numbers leads to a lot of problems. For one thing, infinitesimals can't exist in the usual setting of real numbers! Because the real numbers satisfy an important property, called the Archimedean Property: given any positive real number $\epsilon\gt 0$, no matter how small, and given any positive real number $M\gt 0$, no matter how big, there exists a natural number $n$ such that $n\epsilon\gt M$. But an "infinitesimal" $\xi$ is supposed to be so small that no matter how many times you add it to itself, it never gets to $1$, contradicting the Archimedean Property. Other problems: Leibniz defined the tangent to the graph of $y=f(x)$ at $x=a$ by saying "Take the point $(a,f(a))$; then add an infinitesimal amount to $a$, $a+dx$, and take the point $(a+dx,f(a+dx))$, and draw the line through those two points." But if they are two different points on the graph, then it's not a tangent, and if it's just one point, then you can't define the line because you just have one point. That's just two of the problems with infinitesimals. (See below where it says "However...", though.)

So Calculus was essentially rewritten from the ground up in the following 200 years to avoid these problems, and you are seeing the results of that rewriting (that's where limits came from, for instance). Because of that rewriting, the derivative is no longer a quotient, now it's a limit: $$\lim_{h\to0 }\frac{f(x+h)-f(x)}{h}.$$ And because we cannot express this limit-of-a-quotient as a-quotient-of-the-limits (both numerator and denominator go to zero), then the derivative is not a quotient.

However, Leibniz's notation is very suggestive and very useful; even though derivatives are not really quotients, in many ways they behave as if they were quotients. So we have the Chain Rule: $$\frac{dy}{dx} = \frac{dy}{du}\;\frac{du}{dx}$$ which looks very natural if you think of the derivatives as "fractions". You have the Inverse Function theorem, which tells you that $$\frac{dx}{dy} = \frac{1}{\quad\frac{dy}{dx}\quad},$$ which is again almost "obvious" if you think of the derivatives as fractions. So, because the notation is so nice and so suggestive, we keep the notation even though the notation no longer represents an actual quotient, it now represents a single limit. In fact, Leibniz's notation is so good, so superior to the prime notation and to Newton's notation, that England fell behind all of Europe for centuries in mathematics and science because, due to the fight between Newton's and Leibniz's camp over who had invented Calculus and who stole it from whom (consensus is that they each discovered it independently), England's scientific establishment decided to ignore what was being done in Europe with Leibniz notation and stuck to Newton's... and got stuck in the mud in large part because of it.

(Differentials are part of this same issue: originally, $dy$ and $dx$ really did mean the same thing as those symbols do in $\frac{dy}{dx}$, but that leads to all sorts of logical problems, so they no longer mean the same thing, even though they behave as if they did.)

So, even though we write $\frac{dy}{dx}$ as if it were a fraction, and many computations look like we are working with it like a fraction, it isn't really a fraction (it just plays one on television).

However... There is a way of getting around the logical difficulties with infinitesimals; this is called nonstandard analysis. It's pretty difficult to explain how one sets it up, but you can think of it as creating two classes of real numbers: the ones you are familiar with, that satisfy things like the Archimedean Property, the Supremum Property, and so on, and then you add another, separate class of real numbers that includes infinitesimals and a bunch of other things. If you do that, then you can, if you are careful, define derivatives exactly like Leibniz, in terms of infinitesimals and actual quotients; if you do that, then all the rules of Calculus that make use of $\frac{dy}{dx}$ as if it were a fraction are justified because, in that setting, it is a fraction. Still, one has to be careful because you have to keep infinitesimals and regular real numbers separate and not let them get confused, or you can run into some serious problems.

• As a physicist, I prefer Leibniz notation simply because it is dimensionally correct regardless of whether it is derived from the limit or from nonstandard analysis. With Newtonian notation, you cannot automatically tell what the units of $y'$ are. Commented Mar 10, 2011 at 16:34
• Have you any evidence for your claim that "England fell behind Europe for centuries"? Commented Mar 21, 2011 at 22:05
• @Kevin: Look at the history of math. Shortly after Newton and his students (Maclaurin, Taylor), all the developments in mathematics came from the Continent. It was the Bernoullis, Euler, who developed Calculus, not the British. It wasn't until Hamilton that they started coming back, and when they reformed math teaching in Oxford and Cambridge, they adopted the continental ideas and notation. Commented Mar 22, 2011 at 1:42
• Mathematics really did not have a firm hold in England. It was the Physics of Newton that was admired. Unlike in parts of the Continent, mathematics was not thought of as a serious calling. So the "best" people did other things. Commented Jun 20, 2011 at 19:02
• There's a free calculus textbook for beginning calculus students based on the nonstandard analysis approach here. Also there is a monograph on infinitesimal calculus aimed at mathematicians and at instructors who might be using the aforementioned book.
– tzs
Commented Jul 6, 2011 at 16:24

Just to add some variety to the list of answers, I'm going to go against the grain here and say that you can, in an albeit silly way, interpret $$dy/dx$$ as a ratio of real numbers.

For every (differentiable) function $$f$$, we can define a function $$df(x; dx)$$ of two real variables $$x$$ and $$dx$$ via $$df(x; dx) = f'(x)\,dx.$$ Here, $$dx$$ is just a real number, and no more. (In particular, it is not a differential 1-form, nor an infinitesimal.) So, when $$dx \neq 0$$, we can write: $$\frac{df(x;dx)}{dx} = f'(x).$$

All of this, however, should come with a few remarks.

It is clear that these notations above do not constitute a definition of the derivative of $$f$$. Indeed, we needed to know what the derivative $$f'$$ meant before defining the function $$df$$. So in some sense, it's just a clever choice of notation.

But if it's just a trick of notation, why do I mention it at all? The reason is that in higher dimensions, the function $$df(x; dx)$$ actually becomes the focus of study, in part because it contains information about all the partial derivatives.

To be more concrete, for multivariable functions $$f\colon R^n \to R$$, we can define a function $$df(x;dx)$$ of two n-dimensional variables $$x, dx \in R^n$$ via $$df(x;dx) = df(x_1,\ldots,x_n; dx_1, \ldots, dx_n) = \frac{\partial f}{\partial x_1}dx_1 + \ldots + \frac{\partial f}{\partial x_n}dx_n.$$

Notice that this map $$df$$ is linear in the variable $$dx$$. That is, we can write: $$df(x;dx) = (\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}) \begin{pmatrix} dx_1 \\ \vdots \\ dx_n \\ \end{pmatrix} = A(dx),$$ where $$A$$ is the $$1\times n$$ row matrix of partial derivatives.

In other words, the function $$df(x; dx)$$ can be thought of as a linear function of $$dx$$, whose matrix has variable coefficients (depending on $$x$$).

So for the $$1$$-dimensional case, what is really going on is a trick of dimension. That is, we have the variable $$1\times1$$ matrix ($$f'(x)$$) acting on the vector $$dx \in R^1$$ -- and it just so happens that vectors in $$R^1$$ can be identified with scalars, and so can be divided.

Finally, I should mention that, as long as we are thinking of $$dx$$ as a real number, mathematicians multiply and divide by $$dx$$ all the time -- it's just that they'll usually use another notation. The letter "$$h$$" is often used in this context, so we usually write $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h},$$ rather than, say, $$f'(x) = \lim_{dx \to 0} \frac{f(x+dx) - f(x)}{dx}.$$ My guess is that the main aversion to writing $$dx$$ is that it conflicts with our notation for differential $$1$$-forms.

EDIT: Just to be even more technical, and at the risk of being confusing to some, we really shouldn't even be regarding $$dx$$ as an element of $$R^n$$, but rather as an element of the tangent space $$T_xR^n$$. Again, it just so happens that we have a canonical identification between $$T_xR^n$$ and $$R^n$$ which makes all of the above okay, but I like the distinction between tangent space and Euclidean space because it highlights the different roles played by $$x \in R^n$$ and $$dx \in T_xR^n$$.

• Cotangent space. Also, in case of multiple variables if you fix $dx^2,\ldots,dx^n = 0$ you can still divide by $dx^1$ and get the derivative. And nothing stops you from defining differential first and then defining derivatives as its coefficients. Commented Feb 11, 2011 at 2:30
• Well, canonically differentials are members of contangent bundle, and $dx$ is in this case its basis. Commented Feb 11, 2011 at 5:54
• Maybe I'm misunderstanding you, but I never made any reference to differentials in my post. My point is that $df(x;dx)$ can be likened to the pushforward map $f_*$. Of course one can also make an analogy with the actual differential 1-form $df$, but that's something separate. Commented Feb 11, 2011 at 6:18
• Sure the input is a vector, that's why these linearizations are called covectors, which are members of cotangent space. I can't see why you are bringing up pushforwards when there is a better description right there. Commented Feb 11, 2011 at 9:12
• I just don't think pushforward is the best way to view differential of a function with codomain $\mathbb{R}$ (although it is perfectly correct), it's just a too complex idea that has more natural treatment. Commented Feb 12, 2011 at 4:25

My favorite "counterexample" to the derivative acting like a ratio: the implicit differentiation formula for two variables. We have $$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}$$

The formula is almost what you would expect, except for that pesky minus sign.

See Implicit differentiation for the rigorous definition of this formula.

• Yes, but their is a fake proof of this that comes from that kind a reasoning. If $f(x,y)$ is a function of two variables, then $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}$. Now if we pick a level curve $f(x,y)=0$, then $df=0$, so solving for $\frac{dy}{dx}$ gives us the expression above. Commented Jun 16, 2013 at 19:03
• Pardon me, but how is this a "fake proof"? Commented Apr 28, 2014 at 0:04
• @Lurco: He meant $df = \frac{∂f}{∂x} dx + \frac{∂f}{∂y} dy$, where those 'infinitesimals' are not proper numbers and hence it is wrong to simply substitute $df=0$, because in fact if we are consistent in our interpretation $df=0$ would imply $dx=dy=0$ and hence we can't get $\frac{dy}{dx}$ anyway. But if we are inconsistent, we can ignore that and proceed to get the desired expression. Correct answer but fake proof. Commented May 13, 2014 at 9:58
• I agree that this is a good example to show why such notation is not so simple as one might think, but in this case I could say that $dx$ is not the same as $∂x$. Do you have an example where the terms really cancel to give the wrong answer? Commented May 13, 2014 at 10:07
• @JohnRobertson of course it is. That's the whole point here, that this sort of "fake proof" thinking leads to wrong results here. I blatantly ignore the fact that $d \neq \partial$ (essentially), and I also completely ignore what $F$ really is. My only point here is that if you try to use this as a mnemonic (or worse, a "proof" method), you will get completely wrong results. Commented May 19, 2015 at 18:58

It is best to think of $\frac{d}{dx}$ as an operator which takes the derivative, with respect to $x$, of whatever expression follows.

• This is an opinion offered without any justification.
– user13618
Commented Apr 30, 2014 at 5:20
• What kind of justification do you want? It is a very good argument for telling that $\frac{dy}{dx}$ is not a fraction! This tells us that $\frac{dy}{dx}$ needs to be seen as $\frac{d}{dx}(y)$ where $\frac{d}{dx}$ is an opeartor.
– Emo
Commented May 2, 2014 at 20:43
• Over the hyperreals, $\frac{dy}{dx}$ is a ratio and one can view $\frac{d}{dx}$ as an operator. Therefore Tobin's reply is not a good argument for "telling that dy/dx is not a fraction". Commented Dec 15, 2014 at 16:11
• This doesn't address the question. $\frac{y}{x}$ is clearly a ratio, but it can be also thought as the operator $\frac{1}{x}$ acting on $y$ by multiplication, so "operator" and "ratio" are not exclusive. Commented Jan 29, 2019 at 10:52
• Thanks for the comments - I think all of these criticisms of my answer are valid. Commented Jan 30, 2019 at 19:38

In Leibniz's mathematics, if $y=x^2$ then $\frac{dy}{dx}$ would be "equal" to $2x$, but the meaning of "equality" to Leibniz was not the same as it is to us. He emphasized repeatedly (for example in his 1695 response to Nieuwentijt) that he was working with a generalized notion of equality "up to" a negligible term. Also, Leibniz used several different pieces of notation for "equality". One of them was the symbol "$\,{}_{\ulcorner\!\urcorner}\,$". To emphasize the point, one could write $$y=x^2\quad \rightarrow \quad \frac{dy}{dx}\,{}_{\ulcorner\!\urcorner}\,2x$$ where $\frac{dy}{dx}$ is literally a ratio. When one expresses Leibniz's insight in this fashion, one is less tempted to commit an ahistorical error of accusing him of having committed a logical inaccuracy.

In more detail, $\frac{dy}{dx}$ is a true ratio in the following sense. We choose an infinitesimal $\Delta x$, and consider the corresponding $y$-increment $\Delta y = f(x+\Delta x)-f(x)$. The ratio $\frac{\Delta y}{\Delta x}$ is then infinitely close to the derivative $f'(x)$. We then set $dx=\Delta x$ and $dy=f'(x)dx$ so that $f'(x)=\frac{dy}{dx}$ by definition. One of the advantages of this approach is that one obtains an elegant proof of chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ by applying the standard part function to the equality $\frac{\Delta y}{\Delta x}=\frac{\Delta y}{\Delta u}\frac{\Delta u}{\Delta x}$.

In the real-based approach to the calculus, there are no infinitesimals and therefore it is impossible to interpret $\frac{dy}{dx}$ as a true ratio. Therefore claims to that effect have to be relativized modulo anti-infinitesimal foundational commitments.

Note 1. I recently noticed that Leibniz's $\,{}_{\ulcorner\!\urcorner}\,$ notation occurs several times in Margaret Baron's book The origins of infinitesimal calculus, starting on page 282. It's well worth a look.

Note 2. It should be clear that Leibniz did view $\frac{dy}{dx}$ as a ratio. (Some of the other answers seem to be worded ambiguously with regard to this point.)

• This is somewhat beside the point, but I don't think that applying the standard part function to prove the Chain Rule is particularly more (or less) elegant than applying the limit as $\Delta{x} \to 0$. Both attempts hit a snag since $\Delta{u}$ might be $0$ when $\Delta{x}$ is not (regardless of whether one is thinking of $\Delta{x}$ as an infinitesimal quantity or as a standard variable approaching $0$), as for example when $u = x \sin(1/x)$. Commented Feb 21, 2018 at 23:26
• This snag does exist in the epsilon-delta setting, but it does not exist in the infinitesimal setting because if the derivative is nonzero then one necessarily has $\Delta u\not=0$, and if the derivative is zero then there is nothing to prove. @TobyBartels Commented Feb 22, 2018 at 9:47
• Notice that the function you mentioned is undefined (or not differentiable if you define it) at zero, so chain rule does not apply in this case anyway. @TobyBartels Commented Feb 22, 2018 at 10:15
• Sorry, that should be $u = x^2 \sin(1/x)$ (extended by continuity to $x = 0$, which is the argument at issue). If the infinitesimal $\Delta{x}$ is $1/(n\pi)$ for some (necessarily infinite) hyperinteger $n$, then $\Delta{u}$ is $0$. It's true that in this case, the derivative $\mathrm{d}u/\mathrm{d}x$ is $0$ too, but I don't see why that matters; why is there nothing to prove in that case? (Conversely, if there's nothing to prove in that case, then doesn't that save the epsilontic proof as well? That's the only way that $\Delta{u}$ can be $0$ arbitrarily close to the argument.) Commented Feb 23, 2018 at 12:21
• If $\Delta u$ is zero then obviously $\Delta y$ is also zero and therefore both sides of the formula for chain rule are zero. On the other hand, if the derivative of $u=g(x)$ is nonzero then $\Delta u$ is necessarily nonzero. This is not necessarily the case when one works with finite differences. @TobyBartels Commented Feb 24, 2018 at 19:28

Typically, the $\frac{dy}{dx}$ notation is used to denote the derivative, which is defined as the limit we all know and love (see Arturo Magidin's answer). However, when working with differentials, one can interpret $\frac{dy}{dx}$ as a genuine ratio of two fixed quantities.

Draw a graph of some smooth function $f$ and its tangent line at $x=a$. Starting from the point $(a, f(a))$, move $dx$ units right along the tangent line (not along the graph of $f$). Let $dy$ be the corresponding change in $y$.

So, we moved $dx$ units right, $dy$ units up, and stayed on the tangent line. Therefore the slope of the tangent line is exactly $\frac{dy}{dx}$. However, the slope of the tangent at $x=a$ is also given by $f'(a)$, hence the equation

$$\frac{dy}{dx} = f'(a)$$

holds when $dy$ and $dx$ are interpreted as fixed, finite changes in the two variables $x$ and $y$. In this context, we are not taking a limit on the left hand side of this equation, and $\frac{dy}{dx}$ is a genuine ratio of two fixed quantities. This is why we can then write $dy = f'(a) dx$.

• This sounds a lot like the explanation of differentials that I recall hearing from my Calculus I instructor (an analyst of note, an expert on Wiener integrals): "$dy$ and $dx$ are any two numbers whose ratio is the derivative . . . they are useful for people who are interested in (sniff) approximations."
– bof
Commented Dec 28, 2013 at 3:34
• @bof: But we can't describe almost every real number in the real world, so I guess having approximations is quite good. =) Commented May 13, 2014 at 10:13
• @user21820 anything that we can approximate to arbitrary precision we can define... It's the result of that algorithm.
– k_g
Commented May 18, 2015 at 1:34
• @k_g: Yes of course. My comment was last year so I don't remember what I meant at that time anymore, but I probably was trying to say that since we already are limited to countably many definable reals, it's much worse if we limit ourselves even further to closed forms of some kind and eschew approximations. Even more so, in the real world we rarely have exact values but just confidence intervals anyway, and so approximations are sufficient for almost all practical purposes. Commented May 18, 2015 at 5:04
• This is very close to the way Vladimir Arnol'd explains why the derivative is really a ratio of real numbers in his work on Ordinary Differential Equations. Commented Apr 6 at 4:28

Of course it is a ratio.

$dy$ and $dx$ are differentials. Thus they act on tangent vectors, not on points. That is, they are functions on the tangent manifold that are linear on each fiber. On the tangent manifold the ratio of the two differentials $\frac{dy}{dx}$ is just a ratio of two functions and is constant on every fiber (except being ill defined on the zero section) Therefore it descends to a well defined function on the base manifold. We refer to that function as the derivative.

As pointed out in the original question many calculus one books these days even try to define differentials loosely and at least informally point out that for differentials $dy = f'(x) dx$ (Note that both sides of this equation act on vectors, not on points). Both $dy$ and $dx$ are perfectly well defined functions on vectors and their ratio is therefore a perfectly meaningful function on vectors. Since it is constant on fibers (minus the zero section), then that well defined ratio descends to a function on the original space.

At worst one could object that the ratio $\frac{dy}{dx}$ is not defined on the zero section.

• Can anything meaningful be made of higher order derivatives in the same way? Commented Mar 8, 2015 at 10:41
• You can simply mimic the procedure to get a second or third derivative. As I recall when I worked that out the same higher partial derivatives get realized in it multiple ways which is awkward. The standard approach is more direct. It is called Jets and there is currently a Wikipedia article on Jet (mathematics). Commented May 2, 2015 at 16:51
• Tangent manifold is the tangent bundle. And what it means is that dy and dx are both perfectly well defined functions on the tangent manifold, so we can divide one by the other giving dy/dx. It turns out that the value of dy/dx on a given tangent vector only depends on the base point of that vector. As its value only depends on the base point, we can take dy/dx as really defining a function on original space. By way of analogy, if f(u,v) = 3*u + sin(u) + 7 then even though f is a function of both u and v, since v doesn't affect the output, we can also consider f to be a function of u alone. Commented Jul 6, 2015 at 15:53
• Your answer is in the opposition with many other answers here! :) I am confused! So is it a ratio or not or both!? Commented Jul 15, 2016 at 20:24
• How do you simplify all this to the more special-case level of basic calculus where all spaces are Euclidean? The invocations of manifold theory suggest this is an approach that is designed for non-Euclidean geometries. Commented Feb 3, 2017 at 7:19

The notation $dy/dx$ - in elementary calculus - is simply that: notation to denote the derivative of, in this case, $y$ w.r.t. $x$. (In this case $f'(x)$ is another notation to express essentially the same thing, i.e. $df(x)/dx$ where $f(x)$ signifies the function $f$ w.r.t. the dependent variable $x$. According to what you've written above, $f(x)$ is the function which takes values in the target space $y$).

Furthermore, by definition, $dy/dx$ at a specific point $x_0$ within the domain $x$ is the real number $L$, if it exists. Otherwise, if no such number exists, then the function $f(x)$ does not have a derivative at the point in question, (i.e. in our case $x_0$).

For further information you can read the Wikipedia article: http://en.wikipedia.org/wiki/Derivative

• So glad that wikipedia finally added an entry for the derivative...  Commented Aug 10, 2011 at 17:50

$$\boldsymbol{\dfrac{dy}{dx}}$$ is definitely not a ratio - it is the limit (if it exists) of a ratio. This is Leibniz's notation of the derivative (c. 1670) which prevailed to the one of Newton $$\dot{y}(x)$$.

Still, most Engineers and even many Applied Mathematicians treat it as a ratio. A very common such case is when solving separable ODEs, i.e. equations of the form $$\frac{dy}{dx}=f(x)g(y),$$ writing the above as $$f(x)\,dx=\frac{dy}{g(y)},$$ and then integrating.

Apparently this is not Mathematics, it is a symbolic calculus.

Why are we allowed to integrate the left hand side with respect to to $$x$$ and the right hand side with respect to to $$y$$? What is the meaning of that?

This procedure often leads to the right solution, but not always. For example applying this method to the IVP $$\frac{dy}{dx}=y+1, \quad y(0)=-1,\qquad (\star)$$ we get, for some constant $$c$$, $$\ln (y+1)=\int\frac{dy}{y+1} = \int dx = x+c,$$ equivalently $$y(x)=\mathrm{e}^{x+c}-1.$$ Note that it is impossible to incorporate the initial condition $$y(0)=-1$$, as $$\mathrm{e}^{x+c}$$ never vanishes. By the way, the solution of $$(\star)$$ is $$y(x)\equiv -1$$.

Even worse, consider the case of the IVP $$y'=\frac{3y^{1/3}}{2}, \quad y(0)=0. \tag{\star\star}$$ This IVP does not enjoy uniqueness. It possesses infinitely many solutions. Nevertheless, using this symbolic calculus, we obtain that $$y^{2/3}=x$$, which is one of the infinitely many solutions of $$(\star\star)$$. Another one is $$y\equiv 0$$.

In my opinion, Calculus should be taught rigorously, with $$\delta$$'s and $$\varepsilon$$'s. Once these are well understood, then one can use such symbolic calculus, provided that he/she is convinced under which restrictions it is indeed permitted.

• I would disagree with this to some extent with your example as many would write the solution as $y(x)=e^{x+c}\rightarrow y(x)=e^Ce^x\rightarrow y(x)=Ce^x$ for the 'appropriate' $C$. Then we have $y(0)=Ce-1=-1$, implying $C=0$ avoiding the issue which is how many introductory D.E. students would answer the question so the issue is never noticed. But yes, $\frac{dy}{dx}$ is certainly not a ratio. Commented Dec 20, 2013 at 21:11
• Your example works if $dy/dx$ is handled naively as a quotient. Given $dy/dx = y+1$, we can deduce $dx = dy/(y+1)$, but as even undergraduates know, you can't divide by zero, so this is true only as long as $y+1 \ne 0$. Thus we correctly conclude that $(\star)$ has no solution such that $y+1 \ne 0$. Solving for $y+1=0$, we have $dy/dx = 0$, so $y = \int 0 dx = 0 + C$, and $y(0)=-1$ constraints $C=-1$. Commented Aug 28, 2014 at 14:47
• Since you mention Leibniz, it may be helpful to clarify that Leibniz did view $\frac{dy}{dx}$ as a ratio, for the sake of historical accuracy. Commented Dec 7, 2015 at 18:59
• +1 for the interesting IVP example, I have never noticed that subtlety. Commented Apr 22, 2017 at 3:04
• You got the wrong answer because you divided by zero, not because there's anything wrong with treating the derivative as a ratio. Commented Apr 29, 2017 at 4:20

It is not a ratio, just as $dx$ is not a product.

• I wonder what motivated the downvote. I do find strange that students tend to confuse Leibniz's notation with a quotient, and not $dx$ (or even $\log$!) with a product: they are both indivisible notations... My answer above just makes this point. Commented Feb 10, 2011 at 0:12
• I think that the reason why this confusion arises in some students may be related to the way in which this notation is used for instance when calculating integrals. Even though as you say, they are indivisible, they are separated "formally" in any calculus course in order to aid in the computation of integrals. I suppose that if the letters in $\log$ where separated in a similar way, the students would probably make the same mistake of assuming it is a product. Commented Feb 10, 2011 at 4:09
• I once heard a story of a university applicant, who was asked at interview to find $dy/dx$, didn't understand the question, no matter how the interviewer phrased it. It was only after the interview wrote it out that the student promptly informed the interviewer that the two $d$'s cancelled and he was in fact mistaken. Commented Jan 30, 2012 at 19:54
• Is this an answer??? Or just an imposition? Commented Sep 12, 2013 at 13:12
• I find the statement that "students tend to confuse Leibniz's notation with a quotient" a bit problematic. The reason for this is that Leibniz certainly thought of $\frac{dy}{dx}$ as a quotient. Since it behaves as a ratio in many contexts (such as the chain rule), it may be more helpful to the student to point out that in fact the derivative can be said to be "equal" to the ratio $\frac{dy}{dx}$ if "equality" is interpreted as a more general relation of equality "up to an infinitesimal term", which is how Leibniz thought of it. I don't think this is comparable to thinking of "dx" as a product Commented Oct 2, 2013 at 12:57

In most formulations, $$\frac{dx}{dy}$$ can not be interpreted as a ratio, as $$dx$$ and $$dy$$ do not actually exist in them. An exception to this is shown in this book. How it works, as Arturo said, is we allow infinitesimals (by using the hyperreal number system). It is well formulated, and I prefer it to limit notions, as this is how it was invented. Its just that they weren't able to formulate it correctly back then. I will give a slightly simplified example. Let us say you are differentiating $$y=x^2$$. Now let $$dx$$ be a miscellaneous infinitesimals (it is the same no matter which you choose if your function is differentiate-able at that point.) $$dy=(x+dx)^2-x^2$$ $$dy=2x\times dx+dx^2$$ Now when we take the ratio, it is: $$\frac{dy}{dx}=2x+dx$$

(Note: Actually,$$\frac{\Delta y}{\Delta x}$$ is what we found in the beginning, and $$dy$$ is defined so that $$\frac{dy}{dx}$$ is $$\frac{\Delta y}{\Delta x}$$ rounded to the nearest real number.)

• So, your example is still incomplete. To complete it, you should either take the limit of $dx\to0$, or take standard part of the RHS if you treat $dx$ as infinitesimal instead of as $\varepsilon$. Commented May 8, 2018 at 19:49
• @Ruslan $dx$ is a value, not a dummy variable in a limit. Commented Jan 21, 2021 at 2:33
• I agree with @Ruslan. $dx$ is not a value. There is no real number representing $dx$; it is an infinitesimal. Likewise, limits really contain no dummy variables either; they are essentially infinitesimals. Whether in standard analysis or nonstandard analysis, you must take the limit of $dx$ (in standard analysis this would be represented as $h$) or else this equality would not be useful in any way since there is no real number to represent $dx$. It is also very important to note, given $y$ is a dependent variable and $x$ is an independent variable, $\Delta y \neq dy$., but $\Delta x = dx$ Commented Apr 21, 2023 at 0:40
• @Shidouu nonstandard analysis is based on the hyperreal number system. $dx$ is not a real number, nor is it a variable for a real number. It is a hyperreal. Commented Apr 21, 2023 at 18:29
• I never said $dx$ was a real number. I said it was an infinitesimal, which is a hyperreal. Commented Apr 22, 2023 at 9:03

$\frac{dy}{dx}$ is not a ratio - it is a symbol used to represent a limit.

• This is one possible view on $\frac{dy}{dx}$, related to the fact that the common number system does not contain infinitesimals, making it impossible to justify this symbol as a ratio in that particular framework. However, Leibniz certainly meant it to be a ratio. Furthermore, it can be justified as a ratio in modern infinitesimal theories, as mentioned in some of the other answers. Commented Nov 17, 2013 at 14:57

I realize this is an old post, but I think it's worth while to point out that in the so-called Quantum Calculus $\frac{dy}{dx}$ $is$ a ratio. The subject $starts$ off immediately by saying this is a ratio, by defining differentials and then calling derivatives a ratio of differentials:

The $q-$differential is defined as

$$d_q f(x) = f(qx) - f(x)$$

and the $h-$differential as $$d_h f(x) = f(x+h) - f(x)$$

It follows that $d_q x = (q-1)x$ and $d_h x = h$.

From here, we go on to define the $q-$derivative and $h-$derivative, respectively:

$$D_q f(x) = \frac{d_q f(x)}{d_q x} = \frac{f(qx) - f(x)}{(q-1)x}$$

$$D_h f(x) = \frac{d_h f(x)}{d_q x} = \frac{f(x+h) - f(x)}{h}$$

Notice that

$$\lim_{q \to 1} D_q f(x) = \lim_{h\to 0} D_h f(x) = \frac{df(x)}{x} \neq \text{a ratio}$$

• I just want to point out that @Yiorgos S. Smyrlis did already state that dy/dx is not a ratio, but a limit of a ratio (if it exists). I only included my response because this subject seems interesting (I don't think many have heard of it) and in this subject we work in the confines of it being a ratio... but certainly the limit is not really a ratio. Commented Dec 28, 2013 at 1:54
• You start out saying that it is a ratio and then end up saying that it is not a ratio. It's interesting that you can define it as a limit of ratios in two different ways, but you've still only given it as a limit of ratios, not as a ratio directly. Commented Apr 29, 2017 at 4:40
• I guess you mean to say that the q-derivative and h-derivative are ratios; that the usual derivative may be recovered as limits of these is secondary to your point. Commented May 2, 2017 at 21:55
• Yes, that is precisely my point. Commented Feb 23, 2018 at 4:04

To ask "Is $\frac{dy}{dx}$ a ratio or isn't it?" is like asking "Is $\sqrt 2$ a number or isn't it?" The answer depends on what you mean by "number". $\sqrt 2$ is not an Integer or a Rational number, so if that's what you mean by "number", then the answer is "No, $\sqrt 2$ is not a number."

However, the Real numbers are an extension of the Rational numbers that includes irrational numbers such as $\sqrt 2$, and so, in this set of numbers, $\sqrt 2$ is a number.

In the same way, a differential such as $dx$ is not a Real number, but it is possible to extend the Real numbers to include infinitesimals, and, if you do that, then $\frac{dy}{dx}$ is truly a ratio.

When a professor tells you that $dx$ by itself is meaningless, or that $\frac{dy}{dx}$ is not a ratio, they are correct, in terms of "normal" number systems such as the Real or Complex systems, which are the number systems typically used in science, engineering and even mathematics. Infinitesimals can be placed on a rigorous footing, but sometimes at the cost of surrendering some important properties of the numbers we rely on for everyday science.

See https://en.wikipedia.org/wiki/Infinitesimal#Number_systems_that_include_infinitesimals for a discussion of number systems that include infinitesimals.

Assuming you're happy with $dy/dx$, when it becomes $\ldots dy$ and $\ldots dx$ it means that it follows that what precedes $dy$ in terms of $y$ is equal to what precedes $dx$ in terms of $x$.

"in terms of" = "with reference to".

That is, if "$a \frac{dy}{dx} = b$", then it follows that "$a$ with reference to $y$ = $b$ with reference to $x$". If the equation has all the terms with $y$ on the left and all with $x$ on the right, then you've got to a good place to continue.

The phrase "it follows that" means you haven't really moved $dx$ as in algebra. It now has a different meaning which is also true.

Anything that can be said in mathematics can be said in at least 3 different ways...all things about derivation/derivatives depend on the meaning that is attached to the word: TANGENT. It is agreed that the derivative is the "gradient function" for tangents (at a point); and spatially (geometrically) the gradient of a tangent is the "ratio" ( "fraction" would be better ) of the y-distance to the x-distance. Similar obscurities occur when "spatial and algebraic" are notationally confused.. some people take the word "vector" to mean a track!

• Accordingto John Robinson (2days ago) vectors... elements(points) of vector spaces are different from points Commented May 3, 2014 at 3:43

There are many answers here, but the simplest seems to be missing. So here it is:

Yes, it is a ratio, for exactly the reason that you said in your question.

• Some other people have already more-or-less given this answer, but then go into more detail about how it fits into tangent spaces in higher dimensions and whatnot. That is all very interesting, of course, but it may give the impression that the development of the derivative as a ratio that appears in the original question is not enough by itself. But it is enough. Commented Apr 29, 2017 at 4:50
• Nonstandard analysis, while providing an interesting perspective and being closer to what Leibniz himself was thinking, is also not necessary for this. The definition of differential that is cited in the question is not infinitesimal, but it still makes the derivative into a ratio of differentials. Commented Apr 29, 2017 at 4:50

I am going to join @Jesse Madnick here, and try to interpret $\frac{dy}{dx}$ as a ratio. The idea is: lets interpret $dx$ and $dy$ as functions on $T\mathbb R^2$, as if they were differential forms. For each tangent vector $v$, set $dx(v):=v(x)$. If we identify $T\mathbb R^2$ with $\mathbb R^4$, we get that $(x,y,dx,dy)$ is just the canonical coordinate system for $\mathbb R^4$. If we exclude the points where $dx=0$, then $\frac{dy}{dx} = 2x$ is a perfectly healthy equation, its solutions form a subset of $\mathbb R^4$.

Let's see if it makes any sense. If we fix $x$ and $y$, the solutions form a straight line through the origin of the tangent space at $(x,y)$, its slope is $2x$. So, the set of all solutions is a distribution, and the integral manifolds happen to be the parabolas $y=x^2+c$. Exactly the solutions of the differential equation that we would write as $\frac{dy}{dx} = 2x$. Of course, we can write it as $dy = 2xdx$ as well. I think this is at least a little bit interesting. Any thoughts?

$$dy/dx$$ is quite possibly the most versatile piece of notation in mathematics. It can be interpreted as

1. A shorthand for the limit of a quotient: $$\frac{dy}{dx}=\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x} \, .$$
2. The result of applying the derivative operator, $$d/dx$$, on a given expression $$y$$.
3. The ratio of two infinitesimals $$dy$$ and $$dx$$ (with this interpretation made rigorous using non-standard analysis).
4. The ratio of two differentials $$dy$$ and $$dx$$ acting along the tangent line to a given curve.

All of these interpretations are equally valid and useful in their own way. In interpretations (1) and (2), which are the most common, $$dy/dx$$ is not seen as a ratio, despite the fact that it often behaves as one. Interpretations (3) and (4) offer viable alternatives. Since Mikhail Katz has already given a good exposition of infinitesimals, let me focus the remainder of this answer on interpretation (4).

Given a curve $$y=f(x)$$, the equation of the line tangent to the point $$(a,f(a))$$ is given by $$g(x)=f'(a)(x-a)+f(a) \, .$$ This tangent line provides us with a linear approximation of a function around a given point. For small $$h$$, the value of $$f(a+h)$$ is roughly equal to $$g(a+h)$$. Hence, $$f(a+h) \approx f(a) + f'(a)h$$ The linear approximation formula has a very clear geometric meaning, where you imagine staying on the tangent line rather than the curve itself:

We can then define $$dx=h$$ and $$dy=g(a+h)-g(a)$$:

Since the tangent line has a constant gradient, we have $$\frac{dy}{dx}=\frac{g(a+h)-g(a)}{h}=g'(a)=f'(a) \, ,$$ provided that $$dx$$ is non-zero. Here, $$dy$$ and $$dx$$ are genuine quantities that act along the tangent line to the curve, and can be manipulated like numbers becasuse they are numbers.

This notion of $$dy/dx$$ becomes very useful when you realise that, in a very meaningful sense, the tangent line is the best linear approximation of a function around a given point. We can turn the linear approximation formula into an exact equality by letting $$r(h)$$ be the remainder term: $$f(a+h)=g(a+h)+r(h)=f(a)+f'(a)h+r(h) \, .$$ As $$h\to0$$, $$r(h)\to0$$. In fact, the remainder term satisifes a stronger condition: $$\lim_{h \to 0}\frac{r(h)}{h}=0 \, .$$ The fact that the above limit is equal to $$0$$ demonstrates just how good the tangent line is at approximating the local behaviour of a function. It's not that impressive that when $$h$$ is small, $$r(h)$$ is also very small (any good approximation should have this property). What makes the tangent line unique is that when $$h$$ is small, $$r(h)$$ is orders of magnitude smaller, meaning that the 'relative error' is tiny. This relative error is showcased in the below animation, where the length of the green line is $$h$$, whereas the length of the blue line is $$r(h)$$:

Seen in this light, the statement $$\frac{dy}{dx}=f'(a)$$ is an expression of equality between the quotient of two numbers $$dy$$ and $$dx$$, and the derivative $$f'(a)$$. It makes perfect sense to multiply both sides by $$dx$$, to get $$dy=f'(a)dx \, ,$$ Note that $$dy+r(h)=f(a+h)-f(a)$$, and so \begin{align} f(a+h)-f(a) - r(h) &= f'(a)h \\ f(a+h) &= f(a) + f'(a)h + r(h) \, , \end{align} which gives us another notion of the derivative, through the lense of linear approximation.

The derivate $$\frac{dy}{dx}$$ is not a ratio, but rather a representation of a ratio within a limit.

Similarly, $$dx$$ is a representation of $$\Delta x$$ inside a limit with interaction. This interaction can be in the form of multiplication, division, etc. with other things inside the same limit.

This interaction inside the limit is what makes the difference. You see, a limit of a ratio is not necessarily the ratio of the limits, and that is one example of why the interaction is considered to be inside the limit. This limit is hidden or left out in the shorthand notation that Liebniz invented.

The simple fact is that most of calculus is a shorthand representation of something else. This shorthand notation allows us to calculate things more quickly and it looks better than what it is actually representative of. The problem comes in when people expect this notation to act like actual maths, which it can't because it is just a representation of actual maths.

So, in order to see the underlying properties of calculus, we always have to convert it to the actual mathematical form and then analyze it from there. Then by memorization of basic properties and combinations of these different properties we can derive even more properties.

The best way to understand $$d$$ is that of being an operator, with a simple rule

$$df(x)=f'(x)dx$$

If you take this definition then $$dy/dx$$ is indeed a ratio as it is stripping $$f'(x)dx$$ of $$dx$$

$$\frac{dy}{dx}=\frac{y'dx}{dx}=y'$$

This is done in the same manner as $$12/3$$ is stripping $$12=4\cdot3$$ of $$3$$

In a certain context, $$\frac{dy}{dx}$$ is a ratio.

$$\frac{dy}{dx}=s$$ means:

Standard calculus

$$\forall \epsilon\ \exists \delta\ \forall dx$$

If $$0<|dx|\leq\delta$$

If $$(x, y) = (x_0, y_0)$$, but $$(x, y)$$ also could have been $$(x_0+dx, y_0+\Delta y)$$

If $$dy = sdx$$

Then $$\left|\frac{\Delta y}{dx}-\frac{dy}{dx}\right|\leq \epsilon$$

Nonstandard calculus

$$\forall dx$$ where $$dx$$ is a nonzero infinitessimal

$$\exists \epsilon$$ where $$\epsilon$$ is infinitessimal

If $$(x, y) = (x_0, y_0)$$, but $$(x, y)$$ also could have been $$(x_0+dx, y_0+\Delta y)$$

If $$dy = sdx$$

Then $$\frac{\Delta y}{dx} - \frac{dy}{dx} = \epsilon$$

In either case, $$dx$$ gets its meaning from the restriction placed on it (which is described using quantifiers), and $$dy$$ gets its meaning from the value of $$s$$ and the restriction placed on $$dx$$.

Therefore, it makes sense to make a statement about $$\frac{dy}{dx}$$ as a ratio if the statement is appropriately quantified and $$dx$$ is appropriately restricted.

Less formally, $$dx$$ is understood as "the amount by which $$x$$ is nudged", $$dx$$ is understood as "the amount by which $$y$$ is nudged on the tangent line", and $$\Delta y$$ is understood as "the amount by which $$y$$ is nudged on the curve". This is a perfectly sensible way to talk about a rough intuition.

Of course it is a fraction in appropriate definition.

Let me add my view of answer, which is updated text from some other question.

Accordingly, for example, Murray H. Protter, Charles B. Jr. Morrey - Intermediate Calculus-(2012) page 231 differential for function $$f:\mathbb{R} \to \mathbb{R}$$ is defined as function of two variables selected in special way by formula: $$df(x)(h)=f'(x)h$$ so it is linear function with respect to $$h$$ approximating $$f$$ in point $$x$$. Also it can be called 1-form.

This is fully rigorous definition, which does not required anything, then definition/existence of derivative. But here is more: if we define differential as existence of linear approximation in point $$x=x_0$$ for which holds $$f(x)-f(x_0) = A(x-x_0) + o(x-x_0), x \to x_0$$ then from this we obtain, that $$f$$ have derivative in point $$x=x_0$$ and $$A=f'(x_0)$$. So existence of derivative and existence of differential are equivalence requirements. Rudin W. - Principles of mathematical analysis-(1976) page 213.

If we use this definition for identity function $$g(x)=x$$, then we obtain $$dg(x)(h)=dx(h)=g'(x)h=h$$ This gives possibility to understand record $$\frac{dy}{dx}=\frac{df}{dx}$$ exactly as usual fraction of differentials and holds equality $$\frac{df(x)}{dx}=f'(x)$$. Exact record is $$\frac{df(x)(h)}{dx(h)}=\frac{f'(x)h}{h}=f'(x)$$.

Let me note, that we are speaking about single variable approach, not multivariable.

I cannot explain why someone assert, that $$\frac{dy}{dx}$$ cannot be understand as fraction - may be lack of knowledge about differential definition? For any case I bring, additionally to above source, list of books where is definition of differential which gives possibility understand fraction in question:

1. James R. Munkres - Analysis on manifolds-(1997) 252-253 p.
2. Vladimir A. Zorich - Mathematical Analysis I- (2016) 176 p.
3. Loring W. Tu (auth.) - An introduction to manifolds-(2011) 34 p.
4. Herbert Amann, Joachim Escher - Analysis II (v. 2) -(2008) 38 p.
5. Robert Creighton Buck, Ellen F. Buck - Advanced Calculus-(1978) 343 p.
6. Rudin W. - Principles of mathematical analysis-(1976) 213 p.
7. Fichtenholz Gr. M - Course of Differential and Integral Calculus vol. 1 2003 240-241 p.
8. Richard Courant - Differential and Integral Calculus, Vol. I, 2nd Edition -Interscience Publishers (1937), page 107
9. John M.H. Olmsted - Advanced calculus-Prentice Hall (1961), page 90.
10. David Guichard - Single and Multivariable Calculus_ Early Transcendentals (2017), page 144
11. Stewart, James - Calculus-Cengage Learning (2016), page 190
12. Differential in Calculus

For complete justice I mention Michael Spivak - Calculus (2008) 155 p. where author is against understanding of fractions, but argument is from kind "it is not, because it cannot be". Spivak one of my most respected and favorite authors, but "Amicus Plato, sed magis amica veritas".

While it can be approximated by a ratio, the derivative itself is actuality the limit of a ratio. This is made more apparent when one moves to partial derivatives and $$\frac{1}{\partial y/ \partial x}\neq\partial x / \partial y$$ The concept of $$dx$$ being an actual number and the derivative being an actual ratio is mainly a heuristic device that can get useful intuitions if used properly.

If I give my answer from an eye of physicist ,then you may think of following-

For a particle moving along $$x$$-axis with variable velocity, We define instantaneous velocity $$v$$ of an object as rate of change of x-coordinate of particle at that instant and since we define "rate of change", therefore it must be equal to total change divided by time taken to bring that change. since we have to calculate the instantaneous velocity. we assume instant to mean " an infinitesimally short interval of time for which particle can be assumed to be moving with constant velocity and denote this infinitesimal time interval by $$dt$$. Now particle cannot go more than an infinitesimal distance $$dx$$ in an infinitesimally small time. therefore we define instantaneous velocity as

$$v=\frac{dx}{dt}$$ i.e. as ratio of two infinitesimal changes.

This also helps us to get correct units for velocity since for change in position it will be $$m$$ and for change in time it will be $$s$$.

While defining pressure at a point, acceleration, momentum, electric current through a cross section etc. we assume ratio of two infinitesimal quantities.

So I think for practical purposes you may assume it to be a ratio but what actually it is has been well clarified in other answers.

Also from my knowledge of mathematics, when I learnt about differentiating a function also provides slope of tangent, I was told that Leibniz assumed the smooth curve of some function to be made of infinite number infinitesimally small lines joined together ,extending any one of them gives tangent to curve and the slope of that infinitesimally small line = $$\frac{dy}{dx}$$ = slope of tangent that we get on extending that line.

Even I learnt that we would need a magnifying glass of "infinite zoom in power" to see those lines which may not be possible in real.

It is incorrect to call it a ratio but it's heuristically handy to visualize it as a ratio where the $$\frac{l}{\delta}$$ talks about the times a certain $$l>0$$ can be divided by saying an exceedingly small $$\delta>0$$ such that there is an exceedingly small remainder. Now if even $$l$$ is exceedingly small then the relative difference of the two changes can be measured using this "ratio". It is essential to understand therefore that the nature of how $$l$$ and $$\delta$$ are changing is inherently different (if they were same the ratio would be one which is the case). The idea of limits, thus, suggests that the "ratio of how the $$l$$ changes with respect to $$\delta$$ is the derivative of $$l$$ w.r.t. $$\delta$$ ".

However, it is important to remember that this can't be a mathematical ratio because ratios are defined for constant fractions whilst $$\frac{d}{dx}$$ is a changing fraction with no concept of a remainder/decimals (or algebraic rationality in general).

You could probably give it meaning as ratios. Starting from first principles, if $$y = f(x)$$, then you would write: $$dy = f'(x) dx,\\ d^2y = f''(x) dx^2 + f'(x) d^2x,\\ d^3y = f'''(x) dx^3 + 3 f''(x) d^2x dx + f'(x) d^3x,\\ ⋯,$$ from which would follow: $$f'(x) = \frac{dy}{dx},\\ f''(x) = \frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2},\\ f'''(x) = \frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2}\frac{d^2x}{dx^2} + 3\frac{dy}{dx}\left(\frac{d^2x}{dx^2}\right)^2 - \frac{dy}{dx}\frac{d^3x}{dx^3},\\ ⋯.$$

Then you'd adopt the axiom:

Linearization Axiom:
$$d^nx = 0$$, for independent variables $$x$$ for all $$n > 1$$.

This treats differential relations, then, as linear tangent relations. It might be possible to give interpretation for $$d^2x$$, $$d^3x$$, and so on, if you want to formalize tangents for non-linear references, like quadratic surfaces; i.e. "rolling tangents".

Under the Linearization Axiom, it then follows that: $$f'(x) = \frac{dy}{dx}, \quad f''(x) = \frac{d^2y}{dx^2}, \quad f'''(x) = \frac{d^3y}{dx^3}, \quad ⋯.$$

For an intepretation of the differentials $$dx$$, then, you can resort to the standard definitions in differential geometry. This also paves the way for partial derivatives ... provided that you write them down correctly! What I mean by that is, if $$z = f(x,y)$$, then the partials should actually be written as: $$\left(\frac{∂z}{∂x}\right)_y = f_x(x,y), \quad \left(\frac{∂z}{∂y}\right)_x = f_y(x,y),$$ with the all the independent variables explicitly listed!

What is that important? Well, consider partial derivatives with respect to variables $$s$$, $$t$$ and $$u$$ that satisfy the relation: $$s = t + u,$$ where any two of the three are taken as the the independent variables. Then $$\left(\frac{∂z}{∂t}\right)_u = \left(\frac{∂z}{∂s}\right)_t + \left(\frac{∂z}{∂t}\right)_s, \quad \left(\frac{∂z}{∂u}\right)_t = \left(\frac{∂z}{∂s}\right)_t.$$

Without the explicit subscript, there would be confusion on $$∂z/∂t$$.

In fact, suppose $$z = f(x,y)$$, but also $$y = g(x)$$. Then $$\frac{dz}{dx} = \left(\frac{∂z}{∂x}\right)_y + \left(\frac{∂z}{∂y}\right)_x\frac{dy}{dx} = f_x(x,g(y)) + f_y(x,g(y))g'(y),$$ so that $$\frac{dy}{dx} = \frac{\frac{dz}{dx} - \left(\frac{∂z}{∂x}\right)_y}{\left(\frac{∂z}{∂y}\right)_x}.$$

So, going back to the example for $$s = t + u$$; if $$z$$ is a function of $$s$$, $$t$$ and $$u$$, then the total differential for $$z$$ is given by: $$dz = \left(\frac{∂z}{∂s}\right)_t ds + \left(\frac{∂z}{∂t}\right)_s dt.$$ Taking the wedge product in Grassmannian algebra, this produces the following two relations: $$dz∧dt = \left(\frac{∂z}{∂s}\right)_t ds∧dt, \quad dz∧ds = \left(\frac{∂z}{∂s}\right)_t dt∧ds,$$ since $$ds∧ds = 0 = dt∧dt, \quad ds∧dt = -dt∧ds.$$ So, we can treat the partials are ratios, too: $$\left(\frac{∂z}{∂s}\right)_t = \frac{dz∧dt}{ds∧dt}, \quad \left(\frac{∂z}{∂t}\right)_s = \frac{dz∧ds}{dt∧ds}.$$

Then \begin{align} \left(\frac{∂z}{∂t}\right)_u &= \frac{dz∧du}{dt∧du} = \frac{dz∧(ds-dt)}{dt∧(ds-dt)} = \frac{dz∧ds-dz∧dt}{dt∧ds} = \frac{dz∧ds}{dt∧ds} + \frac{dz∧dt}{ds∧dt}\\ &= \left(\frac{∂z}{∂t}\right)_s + \left(\frac{∂z}{∂s}\right)_t,\\ \left(\frac{∂z}{∂u}\right)_t &= \frac{dz∧dt}{du∧dt} = \frac{dz∧dt}{(ds-dt)∧dt} = \frac{dz∧dt}{ds∧dt}\\ &= \left(\frac{∂z}{∂s}\right)_t. \end{align}

Returning to the example $$z = f(x,y)$$, $$y = g(x)$$, we ultimately have one independent variable, so we postulate $$dz(dx∧dy) + dx(dy∧dz) + dy(dz∧dx) = 0.$$ Then it follows that: \begin{align} 0 &= \frac{dz}{dx} + \frac{dy∧dz}{dx∧dy} + \frac{dy}{dx}\frac{dz∧dx}{dx∧dy}\\ &= \frac{dz}{dx} - \frac{dz∧dy}{dx∧dy} - \frac{dy}{dx}\frac{dz∧dx}{dy∧dx}\\ &= \frac{dz}{dx} - \left(\frac{∂z}{∂x}\right)_y - \frac{dy}{dx}\left(\frac{∂z}{∂y}\right)_y. \end{align}

If two variables $$w$$ and $$x$$ depend on two other variables $$y$$ and $$z$$, then we can write: $$\frac{∂(w,x)}{∂(y,z)} = \frac{dw∧dx}{dy∧dz}.$$ Then, we postulate the following identity: $$(dw∧dx)(dy∧dz) + (dw∧dy)(dz∧dx) + (dw∧dz)(dx∧dy) = 0.$$ From this, it then follows that: \begin{align} 0 &= \frac{dw∧dx}{dy∧dz} + \frac{dw∧dy}{dz∧dy}\frac{dx∧dz}{dy∧dz} - \frac{dw∧dz}{dy∧dz}\frac{dx∧dy}{dz∧dy}\\ &= \frac{∂(w,x)}{∂(y,z)} + \left(\frac{∂w}{∂z}\right)_y\left(\frac{∂x}{∂y}\right)_z - \left(\frac{∂w}{∂y}\right)_z\left(\frac{∂x}{∂z}\right)_y. \end{align}

Thus: $$\frac{∂(w,x)}{∂(y,z)} = \left(\frac{∂w}{∂y}\right)_z\left(\frac{∂x}{∂z}\right)_y - \left(\frac{∂w}{∂z}\right)_y\left(\frac{∂x}{∂y}\right)_z.$$

Another example: the thermodynamics for gases involve entropy $$S$$, temperature $$T$$, pressure $$P$$ and volume $$V$$, which satisfy the fundamental relation: $$dT∧dS - dP∧dV = 0.$$ You can take any two of the four variables as independent, except $$(T,S)$$ or $$(P,V)$$. For the other four combinations, you have: $$0 = \frac{dT∧dS - dP∧dV}{dT∧dV} = \frac{dS∧dT}{dV∧dT} - \frac{dP∧dV}{dT∧dV} = \left(\frac{∂S}{∂V}\right)_T - \left(\frac{∂P}{∂T}\right)_V,\\ 0 = \frac{dT∧dS - dP∧dV}{dT∧dP} = \frac{dS∧dT}{dP∧dT} + \frac{dV∧dP}{dT∧dP} = \left(\frac{∂S}{∂P}\right)_T + \left(\frac{∂V}{∂T}\right)_P,\\ 0 = \frac{dT∧dS - dP∧dV}{dP∧dS} = \frac{dT∧dS}{dP∧dS} - \frac{dV∧dP}{dS∧dP} = \left(\frac{∂T}{∂P}\right)_S - \left(\frac{∂V}{∂S}\right)_P,\\ 0 = \frac{dT∧dS - dP∧dV}{dV∧dS} = \frac{dT∧dS}{dV∧dS} + \frac{dP∧dV}{dS∧dV} = \left(\frac{∂T}{∂V}\right)_S + \left(\frac{∂P}{∂S}\right)_V.$$

Finally, we move on to three or more independent variables, discussing the case where $$u$$, $$v$$ and $$w$$ are functions of $$x$$, $$y$$ and $$z$$, and define $$\frac{∂(u,v,w)}{∂(x,y,z)} = \frac{du∧dv∧dw}{dx∧dy∧dz}.$$ But things will get a lot more involved, at this point. So, I'll cut it short at two independent variables.