Derivative of Hyperbolic Tangent of a x

Theorem

$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$

Proof

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

\(=\)

\(\ds \sech^2 x\)

Derivative of $\tanh x$

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds a \sech^2 a x\)

Derivative of Function of Constant Multiple

$\blacksquare$

Also see

• Derivative of $\coth a x$