Derivative of Tanh Function

Tanh (Hyperbolic Tangent) Function

${\color{Magenta} g(z)=tanh(z)=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}}$

Derivative of Tanh (Hyperbolic Tangent) Function

$\begin{align*} \frac{d}{dz} \left ( \frac{e^{z}-e^{-z}}{e^{z}+e^{-z}} \right )&= \frac{e^{z}+e^{-z}}{\left ( e^{z}+e^{-z} \right )^2}d\left ( e^{z}-e^{-z} \right )-\frac{e^{z}-e^{-z}}{\left ( e^{z}+e^{-z} \right )^{2}}d\left ( e^{z}+e^{-z} \right )\\ &=\frac{\left ( e^{z}+e^{-z} \right )\left ( e^{z} +e^{-z}\right )}{\left ( e^{z} +e^{-z}\right )^{2}}- \frac{\left ( e^{z}-e^{-z} \right )\left ( e^{z} -e^{-z}\right )}{\left ( e^{z} +e^{-z}\right )^{2}}\\ &=\frac{\left ( e^{z}+e^{-z} \right )^2 - \left ( e^{z} -e^{-z}\right )^2}{\left ( e^{z} +e^{-z}\right )^{2}} \\ &= 1-(\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}})^2 \\ &= 1-tanh(z)^2 \end{align*}$

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Derivative of Tanh Function

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