Derivative of Hyperbolic Secant of a x

Theorem

$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$

Proof

\(\ds \map {\dfrac \d {\d x} } {\sech x}\)

\(=\)

\(\ds -\sech x \tanh x\)

Derivative of $\sech x$

\(\ds \leadsto \ \ \)

\(\ds \map {\dfrac \d {\d x} } {\sech a x}\)

\(=\)

\(\ds -a \sech a x \tanh a x\)

Derivative of Function of Constant Multiple

$\blacksquare$

Also see

• Derivative of $\csch a x$