Does the sum of partial derivatives have any special meaning?

Question

Say I have a function $f(x_1,x_2,...x_n)$ and $f : \mathbb{R}^n \to \mathbb{R}$, and partial derivatives $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2},...\frac{\partial f}{\partial x_n}$. If I add these all up,$\frac{\partial f}{\partial x_1}+ \frac{\partial f}{\partial x_2} + ...=\sum_{i=1}^n\frac{\partial f}{\partial x_i}$, does this have any special meaning with regards to the original function? Is this the total derivative?

Answer 1

No, the total derivative is the linear map $d_xf$. If $f$ is differentiable at $x$, then $\sum_{j=1}^n \frac{\partial f}{\partial x_j}(x)$ is the directional derivative of $f$ along the vector $v=(1,1,\cdots,1)$, i.e. $$\lim_{t\to 0}\frac{f(x+tv)-f(x)}{t}$$

Answer 2

I will add to Gae's answer : formally total derivative of a function $f$ is denoted as $$df=\frac{\partial f}{\partial x_1}dx_1+\cdots+\frac{\partial f}{\partial x_n}dx_n.$$ The expression you have is $df$ when $(dx_1,\cdots,dx_n)$ is given by $(1,\cdots,1)$. In natural language it can be thought as (linear approximation of) difference of $f$ when you move to $(1,\cdots,1)$ from $(0,\cdots,0)$.