\(\displaystyle{f}{1}{\left({x}\right)}=\frac{{1}}{{{e}^{{x}}+{12}}}\times\frac{{d}}{{\left.{d}{x}\right.}}\times{\left({e}^{{x}}+{12}\right)}\)

\(\displaystyle=\frac{{1}}{{{e}^{{x}}+{12}}}\times{\left({e}^{{x}}+{0}\right)}=\frac{{e}^{{x}}}{{{e}^{{x}}+{12}}}\)

(Since \(\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({\ln{{x}}}\right)}=\frac{{1}}{{x}}\)) and (\(\displaystyle{\sin{{c}}}{e}\frac{{d}}{{\left.{d}{x}\right.}}\)(constant) = 0, \(\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({e}^{{x}}\right)}={e}^{{x}}\))

\(\displaystyle=\frac{{1}}{{{e}^{{x}}+{12}}}\times{\left({e}^{{x}}+{0}\right)}=\frac{{e}^{{x}}}{{{e}^{{x}}+{12}}}\)

(Since \(\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({\ln{{x}}}\right)}=\frac{{1}}{{x}}\)) and (\(\displaystyle{\sin{{c}}}{e}\frac{{d}}{{\left.{d}{x}\right.}}\)(constant) = 0, \(\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({e}^{{x}}\right)}={e}^{{x}}\))