Commutation relations of derivatives of fermionic fields from the commutation relations of the original fields

Question

I have a general question regarding such type of calculations, but let me start with a concrete question. Consider the $bc$- free fermion CFT so that $b(z)$ and $c(z)$ are free fermions with OPE,
$$b(z) c(w) \sim \frac{1}{z-w}$$ and thus commutation relations, $$\{b_r, c_s \} = \delta_{r,-s}.$$ One can construct a weight 1 bosonic field, defined by $$J(z)=:b(z)c(z):.$$ The modes of this field have commutation relations, $$[J_m,J_n]= m \delta_{m,-n}.$$ We can naturally construct field given by derivatives of $\partial _z b(z)$ and $\partial _z c(z)$ are there normally ordered products. I am interested in the computing commutation relations, $$[J_m, ( b \partial _z c)_n], [(b \partial_z c)_m, (b \partial_z c)_n], etc \ldots$$ Is there an elegant way to compute such commutation relations from some OPE/CFT techniques insted of brute forcing it using distributivity of super commutators etc.