# Why don't we use "dx" as the limiting variable when teaching the definition of the derivative with respect to x?

For a function $$f(x)$$ that is differentiable at $$x$$, the derivative of $$f$$ with respect to $$x$$ at the point $$x$$ is usually given as $$\frac{df(x)}{dx} = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}.$$

I think a much more natural (and perfectly rigorous) definition is one that more similarly resembles the usual $$\frac{df(x)}{dx}$$, namely $$\frac{df(x)}{dx} = \lim_{dx\to0} \frac{df(x)}{dx},$$ where $$df = f(x + dx) - f(x).$$

This is essentially the same thing, but to me it carries more meaning.

To me, not only does this more clearly show what the derivative means (the gradient of $$f$$ at $$x$$, as opposed to between two points), but it also allows $$dx$$ -- and likewise $$df$$ -- to be understood as variables like any other, rather than some sort of new notation (which is what I and every one of my classmates thought when first taught differentiation in school). It justifies the algebraic manipulation of $$dx$$ as a variable like any other, for example in Leibniz's notation for the chain rule: $$\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}$$. It would no longer require teachers to answer the question, "but... can we do that?" with, "yes, just trust me." (a teacher would likely need to remind students that $$dx$$ approaches zero; algebraic operations are merely applied before the limit is evaluated)

It also lends to easier understanding of the integral by making concrete the idea that $$dx$$ is just another variable: $$\int_a^b f(x)\ dx$$ is just the sum of all values of f(x) times $$dx$$ in the interval $$[a,b]$$. Since multiplication distributes over addition, it allows the above integral to be (perfectly rigorously) written as $$dx \int_a^b f(x),$$ provided it is understood that this represents the sum as $$dx$$ approaches zero.

It also reveals why differentiation and integration are inverse operations: the derivative of $$f$$ is $$f(x)$$ divided by $$dx$$, while the integral of $$f$$ is $$f(x)$$ multiplied by $$dx$$ (roughly speaking). If the integral of a function's derivative is taken: $$\int \frac{df}{dx} \, dx,$$ the $$dx$$s evaluate to $$1$$, which leaves $$\int df,$$ the sum of infinitely many infinitesimal changes in $$f$$, which is just $$f(x)$$.

Anyway, I was wondering why we don't use $$dx$$ as the limiting variable in the derivative definition -- I see no reason why the limiting variable and the variable of differentiation can't be the same thing.

• that is not perfectly rigorous Apr 26, 2016 at 17:16
• The widths of the intervals $\Delta x$ are not assumed to be uniform in the definition of the definite integral. Even if they were, you can't assume that this represents a sum, it doesn't. Even if this was allowed, $dx$ doesn't "approach zero". There is a problem in your understanding in the previous paragraph as well. The definite integral is not "just the sum of all values of f(x) times $dx$". I agree that the concepts being applied are delicate. But the definition of the definite integral requires a two sided convergence to a single value (sometimes called the "pinching theorem"). Apr 26, 2016 at 17:35
• In the Riemann sum for integrals, the height of the rectangles is $f(a + k \Delta x)$ where $k$ is the iterator of the sum, and the width of the rectangles is $\Delta x$ -- this is what I meant by "all the values of f(x) times $dx$" When you say the definite integral doesn't represent a sum, what does it represent? i thought that the antiderivative and the Riemann sum were the same thing (or at least evaluated to the same thing) Apr 26, 2016 at 17:40
• My guess is that $dx$ already means infinitesimal so there is no point to take its limit Jun 30, 2020 at 21:32
• @Nubtom as previously said the width may not be uniform and again it is not a sum nor the derivative is the quotient which you are thinking Feb 18, 2021 at 13:31

It's common to write $\Delta x$ instead of $h$. It would be bad to use $dx$, because then $dx$ would mean two different things.
• Well, we'd be using it to mean both dx in the usual sense and also $\Delta x$. Apr 26, 2016 at 21:29
• But what's wrong with that? To me, if $x$ is the variable being differentiated with respect to, then a small (infinitesimal) change in $x$ -- $dx$ -- is what should be approaching zero. Apr 27, 2016 at 7:25
• @Nubtom It is very bad to have a definition of the form $a:=\Phi(a)$. May 6, 2016 at 17:11
• @bof why would you want to use the same notation for two distinct concepts, namely $\Delta x$ or $h$ which is a real number vs $dx$ which in differential geometry is a 1-form (and even if you want to avoid terminology like "1-form", this is really what's going on). Also, if you were really taught $dx=\Delta x$ then that is just very poor form. It's like saying an apple is the same thing as a tomato, and calling them both by the term "apple" just because they're both red. It's just all sorts of confusing, misleading and wrong. Feb 18, 2021 at 3:46