The significance of partial derivative notation

Question

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: $$\frac{\partial }{\partial x_i}f(x_1,\cdots ,x_i,\cdots ,x_n)$$

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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?

Answer 1

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.

Answer 2

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.