Derivative of Hyperbolic Tangent Function

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Theorem

$\map {D_z} {\tanh z} = \sech^2 z$

where $\tanh$ is the hyperbolic tangent and $\sech$ is the hyperbolic secant.


Proof

\(\displaystyle \map {D_z} {\tanh z}\) \(=\) \(\displaystyle \map {D_z} {\dfrac {\sinh z} {\cosh z} }\) Definition 2 of Hyperbolic Tangent
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\paren {D_z \sinh z} \cosh z - \sinh z \paren {D_z \cosh z} } {\cosh^2 z}\) Quotient Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\cosh^2 z - \sinh z \paren {D_z \cosh z} } {\cosh^2 z}\) Derivative of Hyperbolic Sine Function
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\cosh^2 z - \sinh^2 z} {\cosh^2 z}\) Derivative of Hyperbolic Cosine Function
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {\cosh^2 z}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \) \(=\) \(\displaystyle \sech^2 z\) Definition 2 of Hyperbolic Secant

$\blacksquare$


Also see


Sources