sure, I don't trust operators so forgive me but I have to ask if I can presume that the order of working through such an example might then follows something like this...
k(r) = r^{2}
H(r) = \frac{d}{dr} ( k(r) g(r) ) ------> using the product rule
=> \frac{d}{dr} ( F(r) H(r) ) ------>...