Differentiating hyperbolic functions.

Question

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed.

1. $\sech^{2} x$.

2. $2\bigl(\cosh(2x) - 2\bigr) \sech^{4}(x)$.

Thanks a million!

Answer 1

For the fist you can use chain rule, noting that: $$ y=\mbox{sech}^2x = (\cosh x)^{-2} $$ so you have: $$ y'=-2 (\cosh x)^{-3}\sinh x = -2 \tanh x \mbox{ sech}^2x $$ For the second use product rule and chain rule or, better, note that : $$ \cosh (2x)=\cosh^2 x+\sinh^2 x $$ so you can semplify.