\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Let \((dx^i)\) be the coordinate covector fields for given local coordinates. Then the differential of a smooth \(f\colon M\to \mathbb{R}\) in local coordinates is given by

\begin{equation*} df=\frac{\partial f}{\partial x^i}dx^i. \end{equation*}

That means the components of \(df\) are its partial derivatives.