Hello, nikchic5!
\(\displaystyle \text{1. Find the derivative: }\L\:f(u)\:=\:\frac{e^{^{14u}}}{e^{^{7u}}\,+\,e^{^{-7u}}}\)
Want to impress/scare your teacher? . . . Eliminate the negative exponent.
Multiply top and bottom by \(\displaystyle e^{^{7u}}:\L\;f(u)\:=\:\frac{e^{^{21u}}}{e^{^{14u}}\,+\,1}\)
Quotient Rule: \(\displaystyle \L\;f'(u)\:=\:\frac{(e^{^{14u}}\,+\,1)\cdot21e^{^{21u}}\,-\,e^{^{21u}}\cdot14e^{^{14u}}}{(e^{^{14y}}\,+\,1)^2}\)
\(\displaystyle \L\;\;\;= \;\frac{21e^{^{35u}} \,+\,21e^{^{21u}} \,- \,14e^{^{35u}}}{(e^{^{14u}}\,+\,1)^2} \;= \;\frac{7e^{^{35u}}\,+\,21e^{^{21u}}}{(e^{^{14u}}\,+\,1)^2} \;= \;\frac{7e^{^{21u}}\left(e^{^{14u}}\,+\,3\right)}{(e^{^{14u}}\,+\,1)^2}\)
