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I've seen the formula for differentials alot, namely

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

but what I think when I see this is that someone is manipulating the "formula"

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

When I think of "differential equation", I think of

$$f\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2},\cdots,\frac{d^ny}{dx^n}\right)=0$$

not

$$f(x,y,dy,dx,\cdots)=0$$

I've heard that $\Delta y$ and $\Delta x$ can be approximated by $dy$ and $dx$ (or maybe its the other way around?), but that doesn't make much sense to me. If you replace $dy$ and $dx$ by $\Delta y$ and $\Delta x$, you sort of have Euler's method, but this still doesn't clear much up for me. So,

What exactly is a differential, and why is it useful?

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5 Answers 5

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The differential of a function $f$ at $x_0$ is simply the linear function which produces the best linear approximation of $f(x)$ in a neighbourhood of $x_0$.

Specifically, among the linear functions $l$ that take the value $f(x_0)$ at $x_0$, there exists at most one such that, in a neighbourhood of $x_0$, we have: $$f(x_0+h)=f(x_0)+l(h)+o(h)$$ It is the linear map $h\longmapsto l(h)$ that we call the differential of $f$ at $x_0$ and denote $\mathrm d\mkern 1mu f_{x_0}$.

In particular, if $f$ is the identity function, it is its own linear approximation, i.e. if we make the usual abuse of language that consists in confusing a function and its value at a generic point, we will write $\mathrm d\mkern 1mu\operatorname{id}=\mathrm d\mkern 1mu x$. Strictly speaking it is the identity map $h \longmapsto h$, but still confusing the function and its formula, we will write $h=\mathrm d\mkern 1mu x$ and finally: $$\mathrm d\mkern 1mu f_{x_0}=f'(x_0)\,\mathrm d\mkern 1mu x$$ thereby meaning the linear approximation: $\;h\longmapsto f'(x_0) h\;$ of $\;h\longmapsto f(x_0+h)-f(x_0)$.

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The right question is not "What is a differential?" but "How do differentials behave?".

Let me explain this by way of an analogy. Suppose I teach you all the rules for adding and multiplying rational numbers. Then you ask me "But what are the rational numbers?"

The answer is: They are anything that obeys those rules. Now in order for that to make sense, we have to know that there's at least one thing that obeys those rules.

So mathematicians solve that problem as follows: First they define ordered pairs of integers. Then they define two ordered pairs $(a,b)$ and $(c,d)$ to be equivalent if $ad=bc$. Then they define an equivalence class to be any set of ordered pairs, all of which are equivalent to each other, and none of which is equivalent to anything outside that set. Then they define a rational number $a/b$ to be the equivalence class of the ordered pair $(a,b)$, where $a$ is any integer and $b$ is any non-zero integer. Then they describe addition and multiplication of equivalence classes in terms of the underlying ordered pairs. For example, they define $a/b + c/d$ to equal $(ad+bc)/bd$ (remembering that each of these expressions stands for a set of ordered pairs. Then they check that the definition makes sense --- for example, if $a/b=e/f$, then $a/b+c/d$ had better equal $e/f+c/d$, so they check this and a bunch of other properties. Finally, they say: Okay. We've found a structure that obeys all the "rational number" rules, so we know that the rational numbers exist. Now that we know that, we can stop thinking about all that structure and just operate with the rules.

So, nobody ever thinks about the rational number 2/3 as a set of ordered pairs, even though, by the above account, that's what it "is".

Differentials are just like that, except that the construction is considerably more complicated than the construction of rational numbers.

But you started working with rational numbers back in elementary school, long before you knew about how to construct them. All you needed to know were the rules for manipulating them. Much later on, if you were interested, you asked someone "What is a rational number?" and maybe you got an explanation like the one I just gave you.

It's important to know that such a construction exists, because that's what assures that the rules you've been using won't lead to a contradiction. But you really don't need to know the construction in order to learn the rules.

Bottom line: It would have been crazy to wait until you had an explanation at that level before you started doing rational number arithmetic.

If you are interested and curious --- and it sounds like you are --- you will eventually learn how to construct differentials on a rigorous basis. I'm not putting that construction in this answer, because all the details would take too long. But it's analogous to the construction of the rational numbers. And, as in that case, the only purpose of the construction is to show that something satisfies the rules you've been working with all these years, to guarantee that those rules are not self-contradictory.

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The theory that explains details of $\mathrm{d}x$ rigorously and formally is quite abstract and complex and not one that I would recommend trying to grasp whilst still going through calculus or real-analysis. However, a quick read through will provide some nice clarification. To that end, I suggest reading this excellent blog post on math.blogoverflow.com:

More than Infinitesimal: What is “$\mathrm{d}x$”?

that provides an excellent explanation of differentials. It does so by introducing differential forms.

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Here's my earlier answer to a question that is not exactly the same, but it may shed light:

What is $dx$ in integration?

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    $\begingroup$ Your answer doesn't really help except in trivial cases, where you can cross-multiply (or divide) the differentials to make them look like derivatives without having other differentials left over. To see what I mean, try to use your answer to explain the equation $m\,dm = -dm\,v_\mathrm{ex} - dm\,dv$. It doesn't make any sense to someone asking this question. $\endgroup$
    – user541686
    Jul 13, 2015 at 2:04
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$dy$ is supposed to indicate some change in the $y$-variable with respect to some change in the $x$-variable in terms of the tangent line of the graph of the function at a given value of $x$. Intuitively, it means the arbitrarily infinitesimal change in the function corresponding to some arbitrary infinitesimal change in the independent variable.

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