  In this paper we describe all differentiable functions $\varphi,\psi\colon
E\to\mathbb{R}$ satisfying the functional-differential equation
\begin{equation*} [\varphi(y) - \varphi(x)]\psi '\bigl(h(x,y)\bigr) = [\psi(y)
- \psi(x)]\varphi '\bigl(h(x,y)\bigr), \end{equation*} for all $x,y\in E$,
$x<y$, where $E \subseteq \mathbb{R}$ is a nonempty open interval,
$h(\cdot,\cdot)$ is a quasi-arithmetic mean, i.e. $h(x,y)=H^{-1}(\alpha H
(x)+\beta H (y))$, $x,y\in E$, for some differentiable and strictly monotone
function $H\colon E \to H(E)$ and fixed $\alpha, \beta\in (0,1)$ with
$\alpha+\beta=1$.
