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In a calculus textbook I have (Calculus, Stewart), it states that for a differentiable function $y=f(x)$, the differential of the function is defined as $$dy=f'(x) dx.$$

It states that $\Delta x\approx dx$ since $\Delta x$ is small. What I fail to understand is that why $dx$ is classified as a differential, and why in the first place $\Delta x$ is replaced with $dx$ when defining $dy$. The book states that $dy$ represents the change in the linearization of the function, and defines $\Delta y$, given that $$\Delta y=f(x+\Delta x)-f(x)$$ as the change in the value of the function, yet it doesn't state why they replaced $\Delta x$ with $dx$ when defining $dy$. My initial thoughts were that $dx$ must be an infinitesimal, yet the highly voted answer in this question:Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? states that infinitesimals are not logical in standard analysis. Basically, I am confused as to why $\Delta x$ and $dx$ are treated as different concepts, and to me to state that $dx=\Delta x$ simply because $\Delta x$ is small does not seem very mathematically rigorous.

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In calculus textbooks, the concept of differential (of a function) should probably be omitted, since calculus students are not ready to understand its actual meaning. In higher mathematics, the differential (in a linear setting) is correctly defined, but this requires some linear algebra to be appreciated.

It is rather misleading to write $\Delta x \approx dx$ when you can't explain what $dx$ is. I could tell you that $dx$ is the canonical basis of $\mathbb{R}^*$, the dual space of $\mathbb{R}$, or even of $(T_p \mathbb{R})^*$, the dual of the tangent space at the point $p$ to the differentiable manifold $\mathbb{R}$, but I suspect you would not like it.

Therefore here is a more elementary interpretation: if the derivative $f'(x_0)$ is usually defined as a number, you can also see that this number induces a map that sends any real number $h$ to the real number $f'(x_0)h$ (multiplication of numbers). This is the differential of $f$ at $x_0$. Now, you can write $dx$ instead of $h$ (please don't ask why!), and you end up with the calculus differential.

curryage
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Siminore
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  • I thought that the concept of a differential was beyond the scope of an elementary calculus text as it was mentioned in passing only (half a page in a 1200 page text). I suppose I will not spend much time on it for now until it is taught in another course. –  Feb 14 '14 at 15:43
  • One can learn to do algebra rigorously with differentials long before one is ready to encounter ideas like sections of a cotangent bundle or any of the other myriad ways to model them. And the notion has to be taught, because it will be used in other classes (and usually even in introductory calculus itself: e.g. in integration by parts, if not elsewhere). IMO the problem is not with introducing the idea early: it's that it's not really taught, and you get silly descriptions like the one the OP was introduced to. –  Feb 14 '14 at 15:58
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    This is a typical US way of thinking. In Italy we tend to teach mathematical concepts only when they can be rigorously defined. We can manipulate things without knowing what these things are: but is this the best way of teaching and doing mathematics? – Siminore Feb 14 '14 at 16:00
  • And we learn what things "are" by manipulating them so that we can see what we can do with them, and how they can be used to solve problems (especially if they relate to prior ideas we might have). I learned what fractions "are" over a decade before I saw a rigorous definition! While set theoretic models of mathematical concepts are very useful for some purposes, they aren't really what a thing "is". e.g. I don't recall the last time the notion of a dual vector crossed my mind when working with differentials. –  Feb 14 '14 at 16:40
  • Again, I prefer the old fresh approach: Bourbaki, Dieudonné, et cetera. – Siminore Feb 15 '14 at 08:58