Partial derivative in O'Neill's book

Question

May be this is a stupid question but... in Definition $10$ of O'Neill's book Semi-Riemannian Geometry With Applications to Relativity, it is stated that the partial derivative of a function $f\in \mathfrak{F}(M)$ with respect to the coordinate $x^i$ of a coordinate system $\xi = (x^1, \ldots, x^n)$ at $p\in M$ is $$ \frac{\partial f}{\partial x^i}(p) = \frac{\partial (f\circ \xi^{-1})}{\partial u^i} (\xi p) \tag{1}, $$ where $u^1, \ldots, u^n$ are the natural coordinate functions on $\mathbb{R}^n$. My question is, why are the variables $u^i$ in the r.h.s of $(1)$. Shouldn't it be just $$ \frac{\partial f}{\partial x^i}(p) = \frac{\partial (f\circ \xi^{-1})}{\partial x^i} (\xi p) , $$ because, in fact, the function $(f\circ \xi^{-1})$ doesn't depend on the variables $u^i$ but on the variables $x^i$. Is it a typo?