lascieflYr

Answered

2022-11-30

The significance of partial derivative notation

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:

$\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}f({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n})$

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 $\mathrm{\partial}$ have a significant meaning?

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:

$\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}f({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n})$

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 $\mathrm{\partial}$ have a significant meaning?

Answer & Explanation

moralizoL31

Expert

2022-12-01Added 9 answers

Or from a mathematical standpoint:

$\frac{d}{d{x}_{i}}f({x}_{1},\dots ,{x}_{i},\dots ,{x}_{n})=\sum _{j=1}^{n}\frac{\partial}{\partial {x}_{j}}f({x}_{1},\dots ,{x}_{n})\xb7\frac{d{x}_{j}}{d{x}_{i}}$

that is, the partial derivative "binds closer" to $f$ than the total derivative.

$\frac{d}{d{x}_{i}}f({x}_{1},\dots ,{x}_{i},\dots ,{x}_{n})=\sum _{j=1}^{n}\frac{\partial}{\partial {x}_{j}}f({x}_{1},\dots ,{x}_{n})\xb7\frac{d{x}_{j}}{d{x}_{i}}$

that is, the partial derivative "binds closer" to $f$ than the total derivative.

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