Disable kerning for math letters / CM fonts incoherent kerning for df and dj

You can use microtype to remove the ligatures and the kerning for this encoding:

\documentclass{article}
\usepackage{microtype}
\DisableLigatures{encoding = OML}
\begin{document}
\loggingoutput
\showoutput
$d f$ \quad $d \strut f$\quad $d j$\quad $d \strut j$
\end{document}

there is kerning between d and a following letter in math mode because don knuth believes that is the way $df$ and similar differential expressions should appear in print. he is adhering to long-standing practice in math publishing, at least in the u.s.

barbara beeton has explained why the “d” is kerned with the following letter. If you're fussy it shouldn't, define your own macro:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\diff}{\mathop{}\!d}
\newcommand{\sdiff}{\mathop{}\!d\/}

\begin{document}

\begin{gather}
\diff f=\int_0^1 f(x)\diff x\\
\sdiff f=\int_0^1 f(x)\sdiff x
\end{gather}

\end{document}

Your preference seems to go to \sdiff (that you can call as you like).