The significance of partial derivative notationIf some function like f depends on just one variable like x, we denote its derivative with respect to the variable by: d d x f ( x )Now if the function happens to depend on n variables we denote its derivative with respect to the ith variable by: ∂ ∂ x i f ( x 1 , ⋯ , x i , ⋯ , 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? : d d x i f ( x 1 , ⋯ , x i , ⋯ , x n )Does the symbol ∂ have a significant meaning?

Question

The significance of partial derivative notationIf some function like   f depends on just one variable like   x, we denote its derivative with respect to the variable by:            d                      d            x        f  (  x  )Now if the function happens to depend on   n variables we denote its derivative with respect to the   ith variable by:      ∂          ∂              x        i              f  (      x    1    ,  ⋯  ,      x    i    ,  ⋯  ,      x    n    )Now my question is what is the significance of this notation? I mean what will be wrong if we show &quot;Partial derivative&quot; of    f with respect to       x    i   like this? :            d                      d                    x        i              f  (      x    1    ,  ⋯  ,      x    i    ,  ⋯  ,      x    n    )Does the symbol   ∂ have a significant meaning?

moralizoL31 · Accepted Answer

Or from a mathematical standpoint:      d          d              x        i              f  (      x    1    ,  …  ,      x    i    ,  …  ,      x    n    )  =      ∑          j      =      1        n              ∂                      ∂                    x        j              f  (      x    1    ,  …  ,      x    n    )  ·            d              x        j                    d              x        i            that is, the partial derivative &quot;binds closer&quot; to   f than the total derivative.