Notation for Total Derivative

Question

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is differentiable, we write its total differential as $$df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx_i.$$ I am a bit confused on what the notation $dx_i$ means in this case? I know when each $x_i$ depends on some $t$ (i.e. $x_i:\mathbb{R}\to\mathbb{R}$), we have $$df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}$$ but the first case confuses me. Wouldn't each $dx_i=1$? If somebody could elaborate on this a bit that would be great.