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If some function like $f$ depends on just one variable like $x$, we denote its derivative with respect to the variable by: $$\frac{\mathrm{d} }{\mathrm{d} x}f(x)$$ Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by: $$\frac{\partial }{\partial x_i}f(x_1,\cdots ,x_i,\cdots ,x_n)$$

Now my question is what is the significance of this notation? I mean what will be wrong if we show "Partial derivative" of $f$ with respect to $x_i$ like this? : $$\frac{\mathrm{d} }{\mathrm{d} x_i}f(x_1,\cdots ,x_i,\cdots ,x_n)$$ Does the symbol $\partial$ have a significant meaning?

Hamed Begloo
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Or from a mathematical point of view: $$ \frac{d}{dx_i}f(x_1,…,x_i,…,x_n)=\sum_{j=1}^n\frac{∂}{∂x_j}f(x_1,…,x_n)·\frac{dx_j}{dx_i} $$ that is, the partial derivative "binds closer" to $f$ than the total derivative.

Lutz Lehmann
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  • Thank you. But keep in mind I just know about elementary calculus :).So my knowledge is not much. What do you mean by "Total derivative"? I mean can you firstly define all of the terms "Ordinary derivative", "Partial derivative" and "Total derivative" in your answer? – Hamed Begloo Nov 05 '16 at 11:32
  • The partial derivative treats all variables as independent. When differentiating for $x_i$, all other $x_j$ are held constant. -- In the total derivative, all variables are considered to be functions of $x_i$, even if they turn out as constant functions. Thus $\frac{d}{dx}f(x,y(x))=f_x+f_y·y'(x)$ is different from $f_x=\frac{∂}{∂x}f(x,y(x))$. – Lutz Lehmann Nov 05 '16 at 11:39
  • So for univariate functions "Ordinary derivative" and "Partial derivative" have the same values. But multivariate functions has no such thing as "Ordinary derivative"(and the notation is used instead for "Total derivative" which means the dependency of variables is not determined). And plus for multivariate functions we have "Partial derivative" which means the other variables are taken as independent. Correct? – Hamed Begloo Nov 05 '16 at 11:52
  • Yes. The boundaries of what is what may be a little fluid from place to place, but essentially this is correct. -- I would not say "have the same values" but "are the same concept". – Lutz Lehmann Nov 05 '16 at 12:56
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Mainly historical; see Earliest Uses of Symbols of Calculus : Partial Derivative :

The "curly $\mathrm{d}$" was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in "Memoire sur les Equations aux différence partielles", which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCLXXIII (1773). On page 152, Condorcet says:

Dans toute la suite de ce Memoire, $\mathrm{d} z$ & $\partial z$ désigneront ou deux differences partielles de $z$, dont une par rapport a $x$, l'autre par rapport a $y$, ou bien $\mathrm{d} z$ sera une différentielle totale, & $\partial z$ une difference partielle. [Throughout this paper, both $\mathrm{d} z$ & $\partial z$ will either denote two partial differences of $z$, where one of them is with respect to $x$, and the other, with respect to $y$, or $\mathrm{d} z$ and $\partial z$ will be employed as symbols of total differential, and of partial difference, respectively.]

However, the "curly $\mathrm{d}$" was first used in the form $\dfrac{\partial u}{\partial x}$ by Adrien Marie Legendre in 1786 in his "Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations", Histoire de l'Academie Royale des Sciences, Annee M. DCCLXXXVI (1786), pp. 7-37, Paris, M. DCCXXXVIII (1788).

On page 8, it reads:

Pour éviter toute ambiguité, je répresentarie par $\dfrac{\partial u}{\partial x}$, le coefficient de $x$ dans la différence de $u$, & par $\dfrac{\mathrm{d} u}{\mathrm{d} x}$ la différence complète de $u$ divisée par $\mathrm{d} x$.

Legendre abandoned the symbol and it was re-introduced by Carl Gustav Jacob Jacobi in 1841.

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Mauro ALLEGRANZA
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  • Sorry but I confused. Does it mean $\partial x$ and $\mathrm{d} x$ have different meanings? Does even $\partial x$ alone have a meaning? – Hamed Begloo Nov 05 '16 at 11:20
  • $\partial x$ doesn't really have any meaning as far as i know. You shouldn't really take these notations apart. $\partial_x$ ($x$ in subscript) is often used as a shorthand for $\frac{\partial}{\partial x}$ – Lord Commander Mar 17 '21 at 23:04