Showing a renormalization result from an introductory paper on Feynman integrals

Question

I am reading a paper on Feynman integrals, where the author states that from the following equation (96), $$Z_g=1-\frac12\beta_0\frac{g_R^2}{(4\pi)^2}+O(g_R^4),$$ we obtain the equation (99) $$Z_g^{-1}\mu\frac{d}{d\mu}Z_g=\beta_0\frac{g_R^2}{(4\pi)^2}+O(g_R^4).$$ Somehow I cannot seem to derive this result. Note that $\beta_0$ is a constant here and $g_R$ is being shown to depend on $\mu$. Previously, it was shown that the equation (98) holds: $$\mu\frac{d}{d\mu}g_R=-\epsilon g_R-(Z_g^{-1}\mu\frac{d}{d\mu}Z_g)g_R$$ and we are taking the limit as $\epsilon\to0$ (it's the dimensional regularization parameter). Can I obtain equation (99) from these two results or am I missing something?