Equations with one catalytic variable and one univariate unkown, also known as discrete difference equations of order one, form a familly of combinatorially relevant functional equations first discussed in full generality by Bousquet-Mélou and Jehanne (2006) who proved that their power serie solutions are algebraic. Drmota, Noy and Yu (2022) recently showed that in the non linear case the singular expansions of these series have a universal dominant term of order 3/2, as opposed to the dominant square root term of generic $\mathbb{N}$-algebraic series. Their direct analysis of the cancellation underlying this behavior is a tour de force of singular analysis. We show that the result can instead be given a straightforward explanation by showing that the derivative of the solution series conforms to the standard square root singular behavior. Consequences also include an atypical, but generic in this situation, $n^{^{5 / 4}}$ asymptotic behavior for the cumulated values of the underlying catalytic parameter.