Why is Leibniz Notation written this way for the second derivative?

Question

Let $y = f(x)$.

$$f'' = \frac{d^2y}{dx^2}$$

$$ \Bigl(\dfrac{d}{dx}\Bigr)^2 y = \dfrac{d^2}{dx^2}\,y = \dfrac{d^2 y}{dx^2};$$

Since there are two $d$'s in the bottom of the fraction, why is it not written

$$\frac{d^2y}{d^2x^2}$$

Maybe it's because $dx$ needs to be thought of as a single thing. But notice that $d$ is used by itself and squared in the numerator..

Does my point here make sense, is it just a convenience to avoid the extra exponent, or is there a logical reason it's written in the form it is?

Perhaps the $dx^2$ is meant to be read as $(dx)^2$ (rather than $d(x^2)$), where one views "$dx$" as one symbol rather than two separate letters. But I am just speculating. — angryavian, Feb 14 '22 at 22:06
$d$ isn’t a number, it more like operator. In particular, $(dx)^2\neq d^2x^2.$ It would be more approapriate to write $$\frac{d^2y}{(dx)^2},$$ but we abandon the standard rules of precedence here, and remove the parentheses. — Thomas Andrews, Feb 14 '22 at 22:11
Duplicate? Why are the $d$'s not altered for higher-order derivatives? — Blue, Feb 14 '22 at 22:14
In functions $f$ of multiple variables, we write $$\frac{d^2f}{dxdy},$$ which shows we are really talking $(dx)^2$ in the denominator, and we really use $dx^2$ as a shorthand. $d$ is not a “number-like” thing, but $dx$ is “number-like.” If $d$ were number-like, we’d get things like $$\frac{dy}{dx}=\frac{y}{x}.$$ That is wrong, and our notation should discourage treating $d$ as a number. — Thomas Andrews, Feb 14 '22 at 22:23
If d is like an operator, how can it be squared in the numerator? — Ben G, Feb 14 '22 at 22:47
See also What does the denominator in the second derivative mean? — Mark S., Feb 15 '22 at 02:19
$d$ is not squared in the numerator. You could view it as an operator applied twice, although I tend to prefer to think of all of $\frac{d}{dx}$ as a differentiation operator and the notation $\frac{d^2}{dx^2}$ as a convenient convention even if it might not be strictly logical (at least when viewed that way). — David K, Feb 15 '22 at 02:48

Masacroso · Accepted Answer · 2022-02-15T06:35:31.180

1

Its customary to write $dx^2$ to denote $(dx)^2$ in all common theories of calculus, then we set

$$ \frac{d}{d x}\circ \frac{d}{d x}=:\left(\frac{d}{d x}\right)^2=:\frac{d^2}{dx^2} $$

The last expression is a whole, that is, $d^2$ and $dx^2$ doesn't make sense by themselves (and the fraction is just a notation resembling the analytic definition of the derivative but it doesn't mean something). The second expression $\left(\frac{d}{d x}\right)^2$ is common for any linear operator to represent the composition of a linear operator with itself, and $\frac{d}{d x}$ is a linear operator in the space of real-to-real smooth functions.

edited Feb 15 '22 at 06:35

answered Feb 15 '22 at 02:21

Masacroso

30,417

Thanks. I don't understand what $d^2$ in the numerator is to be interpreted as.. – Ben G Feb 15 '22 at 04:01
@bgcode the elements alone in this notation have no meaning, its the whole $\frac{d}{d x}$ what makes sense. Then, when we apply this operator twice we use the exponents to denote it. – Masacroso Feb 15 '22 at 05:53
So the $^2$ should be read as "there are two of these" rather than implying anything is multiplied by anything else? – Ben G Feb 15 '22 at 06:21
@bgcode exactly. Mathematicians use many other notations to represent the same that $\frac{d^2}{d x^2}$, by example $D^2$ or $\partial ^2$ are very used (and makes more sense, I mainly use the last one). Think that the notation about you are asking is very old (it was the notation used by Leibniz, one of the creators of calculus!), prior to any formalization of mathematics – Masacroso Feb 15 '22 at 06:23
Ok cool. I've been using it now to solve Parametric Equations (https://www.khanacademy.org/math/ap-calculus-bc/bc-advanced-functions-new/bc-9-2/v/second-derivative-of-parametric-functions).
I would maybe add one line to your answer addressing what we've said in the comment here about it meaning 2 of something not squared, and can grant the answer
– Ben G Feb 15 '22 at 06:31
@bgcode there are many things to say about the Leibniz notation (used differently than in this question), however these things are really advanced, it doesn't belong to a basic course of calculus, it belong to an alternative formalization of real analysis named nonstandard analysis. Also the notation of Leibniz can be, in some sense, be related to the concept of derivation and differential in differential geometry (but as I said, this topic is really advanced) – Masacroso Feb 15 '22 at 06:40
by example, in the nonstandard analysis of Robinson the notation $\frac{dy}{dx}$ represents, literally, a fraction of hyperreal numbers (in particular, a fraction of infinitesimal numbers) and this fraction can be related to the derivative of $y$ (respect to $x$) in a specific way. In differential geometry the notation $\frac{d}{d x}$ represents a kind of vector in a structure named the tangential bundle of $\mathbb{R}$, and applied to real-to-real smooth functions defines it derivative. More to say in this same context is that here $dx$ is the dual vector of $\frac{d}{d x}$, etc... – Masacroso Feb 15 '22 at 06:45
anyway, in the topics mentioned in the previous comments, the components of the notation $\frac{d^2}{d x^2}$ doesn't make sense alone, by example in a hyperreal setting $d^2$ doesn't have a meaning, and in differential geometry the notation is again a shorthand for applying twice the vector (that act as a derivation) $\frac{d}{d x}$ to some smooth function – Masacroso Feb 15 '22 at 06:48

Why is Leibniz Notation written this way for the second derivative?

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