Let y=log(ax+b), then y+dy=log(a(x+dx)+b), where dy is the small change in y for a small change dx in x,
Then dy=y+dy-dy=log(a(x+dx)+b)-log(ax+b)=log((ax+adx+b)/(ax+b))=log(1+adx/(ax+b)).
Natural logs are assumed where log(1+x)=x, to the first approximation. So dy=adx/(ax+b) and dy/dx=a/(ax+b) or a/y.
(This may be a chicken-egg situation, because the power series for ln(1+x) may itself be derived from derivatives and integrals.)