Question

# Find the derivative of the function f(x) = ln(e^x + 12)

Analyzing functions
Find the derivative of the function $$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({e}^{{x}}+{12}\right)}}}$$
$$\displaystyle{f}{1}{\left({x}\right)}=\frac{{1}}{{{e}^{{x}}+{12}}}\times\frac{{d}}{{\left.{d}{x}\right.}}\times{\left({e}^{{x}}+{12}\right)}$$
$$\displaystyle=\frac{{1}}{{{e}^{{x}}+{12}}}\times{\left({e}^{{x}}+{0}\right)}=\frac{{e}^{{x}}}{{{e}^{{x}}+{12}}}$$
(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, $$\displaystyle\frac{{d}}{{\left.{d}{x}\right.}}{\left({e}^{{x}}\right)}={e}^{{x}}$$)