lascieflYr

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:
$\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)$
Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:
$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_{i}}f\left({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n}\right)$
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? :
$\frac{\mathrm{d}}{\mathrm{d}{x}_{i}}f\left({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n}\right)$
Does the symbol $\mathrm{\partial }$ have a significant meaning?

moralizoL31

Expert

Or from a mathematical standpoint:
$\frac{d}{d{x}_{i}}f\left({x}_{1},\dots ,{x}_{i},\dots ,{x}_{n}\right)=\sum _{j=1}^{n}\frac{\partial }{\partial {x}_{j}}f\left({x}_{1},\dots ,{x}_{n}\right)·\frac{d{x}_{j}}{d{x}_{i}}$
that is, the partial derivative "binds closer" to $f$ than the total derivative.

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