If $z=f(x_{1}, x_{2}, \ldots , x_{n})=x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}$, then $$\frac{\partial f}{\partial x_{1}}=2x_{1} \qquad \frac{\partial f}{\partial x_{2}}=2x_{2}\qquad \ldots \qquad \frac{\partial f}{\partial x_{n}}=2x_{n}$$

Collectively, these partial derivatives can be written as $$\frac{\partial f}{\partial x_{i}}=2x_{i} \qquad \hbox{where } i=1,2, \ldots , n$$