Derivative of Hyperbolic Secant Function

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Theorem

$D_z \left({\operatorname{sech} z}\right) = -\operatorname{sech} z \ \tanh z$

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


Proof 1

\(\displaystyle D_z \left({\sech z}\right)\) \(=\) \(\displaystyle D_z \left({\frac 1 {\cosh z} }\right)\) Definition of Hyperbolic Secant
\(\displaystyle \) \(=\) \(\displaystyle D_z \left({\left({\cosh z}\right)^{-1} }\right)\) Exponent Laws
\(\displaystyle \) \(=\) \(\displaystyle -\left({\cosh z}\right)^{-2} \, \sinh z\) Derivative of Hyperbolic Cosine Function, Power Rule for Derivatives, Chain Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1} {\cosh z} \, \frac {\sinh z} {\cosh z}\) Exponent Combination Laws
\(\displaystyle \) \(=\) \(\displaystyle -\sech z \, \tanh z\) Definition of Hyperbolic Secant and Definition of Hyperbolic Tangent

$\blacksquare$


Proof 2

\(\displaystyle \mathrm D_z \left(\operatorname{sech} z \right)\) \(=\) \(\displaystyle 2\mathrm D_z \left(\frac {e^z} {e^{2z} + 1} \right)\) Definition of Hyperbolic Secant
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 {\left( {e^{2z} + 1}\right)^2} \left(\mathrm D_z\left[e^{z}\right]\left[e^{2z} + 1\right] - e^{z}\mathrm D_z\left[e^{2z} + 1\right]\right)\) Quotient Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle - \frac 2 {\left( {e^{2z} + 1}\right)^2} \left(2e^{2z} \cdot e^z - e^{z}\cdot e^{2z} - e^z\right)\) Derivative of Exponential Function
\(\displaystyle \) \(=\) \(\displaystyle - \frac {2\left(e^{3z} - e^{z}\right)} {\left( {e^{2z} + 1} \right)^2}\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {2e^z} {\left( {e^{2z} + 1} \right)} \cdot \frac {e^{2z}-1} {e^{2z} + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle -\operatorname{sech} z \tanh z\) Definition of Hyperbolic Secant, Definition of Hyperbolic Tangent

$\blacksquare$


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