well let's break it down
\(\displaystyle \dfrac{d}{ds}e^s = e^s\)
\(\displaystyle \dfrac{d}{ds} s \ln(s) = \ln(s) + 1\)
\(\displaystyle \dfrac{d}{ds}\dfrac{e^s}{s\ln(s)}= \dfrac{e^s(s \ln(s)) - e^s(\ln(s)+1)}{(s\ln(s))^2} = \\~\\
\dfrac{e^s}{s \ln(s)} - \dfrac{e^s}{s^2\ln(s)} - \dfrac{e^s}{s^2(\ln(s))^2}
\)
you seemed to miss the fact that \(\displaystyle \dfrac{d}{ds} s = 1\)