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derivative notation

This is the list of known standard representations and their nuances.

$\frac{du}{dv}, \frac{df}{dx}, \frac{dy}{dx}-$ The most common notation, this is read as the derivative of $u$ with respect to $v$ . Exponents relate which derivative, for example, $\frac{d^2y}{dx^2}$ is the second derivative of $y$ with respect to $x$ .

$f'(x)\; ,\vec{f}'(\pt{x})\;, y''-$ This is read as $f$ prime of $x$ . The number of primes tells the derivative, ie. $f'''(x)$ is the third derivative of $f(x)$ with respect to $x$ . Note that in higher dimensions, this may be a tensor of a rank equal to the derivative.
$D_xf(\pt{x}), F_y(\pt{x}), f_{xy}(\pt{x})-$ These notations are rather arcane, and should not be used generally, as they have other meanings. For example $F_y$ can easily by the $y$ component of a vector-valued function. The subscript in this case means ``with respect to'', so $F_{yy}$ would be the second derivative of $F$ with respect to $y$ .
$D_1f(\pt{x}), F_2(\pt{x}), f_{12}(\pt{x})-$ The subscripts in these cases refer to the derivative with respect to the nth variable. For example, $F_2(x,y,z)$ would be the derivative of $F$ with respect to $y$ . They can easily represent higher derivatives, ie. $D_{21}f(\pt{x})$ is the derivative with respect to the first variable of the derivative with respect to the second variable.
$\frac{\partial u}{\partial v}\; ,\frac{\partial f}{\partial x}-$ The partial derivative of $u$ with respect to $v$ . This symbol can be manipulated as in $\frac{du}{dv}$ for higher partials.
$\frac{d}{dv}\;,\frac{\partial}{\partial v}-$ This is the operator version of the derivative. Usually you will see it acting on something such as $\frac{d}{dv}(v^2+3u) = 2v$ .
$[\mathbf{Jf}(\pt{x})]\:,[\mathbf{Df}(\pt{x})]-$ The first of these represents the Jacobian of $\mathbf{f}$ , which is a matrix of partial derivatives such that

$\displaystyle [\mathbf{Jf}(\mathbf {x})] = \left[\begin{array}{ccc} D_1f_1(\mat... ...dots\ D_1f_m(\mathbf {x}) & \dots & D_nf_m(\mathbf {x})\ \end{array}\right]$

where $f_n$ represents the nth function of a vector valued function. The second of these notations represents the derivative matrix, which in most cases is the Jacobian, but in some cases, does not exist, even though the Jacobian exists. Note that the directional derivative in the direction $\vec{v}$ is simply $[\mathbf{Jf}(\pt{x})]\vec{v}$ .

derivative notation is owned by Cam McLeman, Ronald Lett.

How to Cite This Entry

Cam McLeman, Ronald Lett. "derivative notation" (version 10). PlanetMath.org. Freely available at http://planetmath.org/DerivativeNotation.html.

Classification

AMS MSC:

26-00 (Real functions :: General reference works (handbooks, dictionaries, bibliographies, etc.))

Pending Errata and Addenda

None. [ View all 5 ]

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Discussion

Higher order (partial) derivatives by Marvin The Martian on 2007-12-12 15:34:29

Hi --

If you forgive me writing in \LaTeX\ (and I hope it compiles), I will want an intuitively-formed (partial) derivative command as such,
%------------------
\newcommand{\pdif}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dif}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}}
%------------------

Then: I am looking for notational conventions for higher order derivatives.

Most notations show what to do for real functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$, interpreting the argument either as a single vector or as $n$ scalars (the second then mentioning that if the intended mixed partial derivative is continuous, then the order doesn't matter). My situation is slightly different, as I'm dealing with $g:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ and similar beasts. If $n \neq m$ then you can infer the order, kind of, but for $n = m$ you have no way of telling.

So for respectively $n$- and $m$-dimensional vectors $X,Y$, how to spell it out? Up to second order works in rowvector-matrix-columnvector, but for third order this approach fails (and the question is: how to denote the matrix?). This issue doesn't appear if you have a single $n+m$-dimensional variable --- we are in essence carving pieces out of this higher-dimensional beast.

My proposal/intuition [1] is:
to evaluate $f(X,Y)$ derived once for $X$ then for $Y$ at $(Xo,Yo)$ is denoted as $\pdif{^2 f}{Y \partial X}[X-Xo,Y-Yo]$. Thus the order of derivation is read from RIGHT to LEFT, and the arguments to evaluate the differential with ($X-Xo$ and $Y-Yo$ here) are added LEFT to RIGHT.
This just generalizes $ g'(x) = \dif{}{x} g(x)$, where $g(x)=\dif{}{x} f(x)$.
Dissatisfaction: this is no stepwise/unambiguous way to get to a matrix notation.

So proposal/intuition [2] is:
the same, but order both what you derive for ($\partial X$ etc) and where you evaluate from LEFT to RIGHT.

If you do this, and agree that deriving for a (column)vector-valued variable $X,Y$ will give us a (ROW)vector-valued object which is just the row vector of derivatives to each of the component variables, then the mentioned mixed derivative is written as a matrix as follows.
$\pdif{^2 f}{X \partial Y} = \pdif{}{Y} [\pdif{f}{X} ]^T$, where $^T$ stands for transpose, which is an $m \times n$ matrix. The advantage is that you have now the intuitive notation $ (X-Xo)^T \pdif{^2 f}{X \partial Y} (Y-Yo)$, so the first variable for deriving ($X$) is next to its argument ($X-Xo$), and the second next to the second.
Dissatisfaction: a first order derivative $\pdif{f}{x} (X-Xo)$ derived again, switches this part from right to left: $ (X-Xo)^T [\pdif{^2 f}{X \partial Y}] (Y-Yo)$.

Proposal/intuition [3] is the same as [2], but transposing the matrix,
(this gives a confusion of mixed effects).

Proposal/intuition [4] is taking the ordering of [1], plus the definition of [2], and (like [3]) then transposes the matrix.
This has both the advantages of [1] and [2] (namely, the derivatives added in the order so that the ``$\dif{^2 f}{x^2} = \dif{}{x} (\dif{f}{x})$" idea fits, and that the argument stands next to its label).

Does this make any sense to you, or have you got any source on this topic? (Incidentally, I have just published a paper using the [2] notation which I regret)

All the best,
M.

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