Question about write differential as sum of partial derivatives

Question

I learned that if i have a function $f:R^p\to R^q$ i can write differential of function like $df(a)(u)=\frac{\partial f}{\partial x_1}(a)u_1+\frac{\partial f}{\partial x_2}(a)u_2+...+\frac{\partial f}{\partial x_p}(a)u_p$, where $u=(u_1,u_2,...,u_p)$ and $pr_i:R\to R, pr_i(u_1,...u_p)=u_i, \forall u\in R^p$. And i saw notations like this: $df(a)(u)=\frac{\partial f}{\partial x_1}(a)dx_1+\frac{\partial f}{\partial x_2}(a)dx_2+...+\frac{\partial f}{\partial x_p}(a)dx_p$.
Why we use this notation and how we get from $u_p \to dx_p$. $dx_p$ mean differential of $x_p$ or is just a notation?