Derivative of the residue logarithm of a formal pseudo-differential series

Question

In my lecture notes in the proof of Adler's theorem they use $$ D_t(res \log A)= res (D_t(A) \circ A^{-1}). $$ Where $D_t$ is derivation of the differential field $F$ and $A$ is a formal pseudo-differential series: $$ A= a_{n} D_x^{n}+a_{n-1} D_x^{n-1}+ ..., \quad a_k \in F. $$ Is this true for all formal pseudo-differential series and derivations? How would you prove this?