# Derivative of Tanh The tanh function is defined as $ \begin{align} \tanh (x)=\frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{sinh(x)}{cosh(x)} \end{align} $ From quotient rule: $ \frac{d(f / g)}{d x}=\frac{g f^{\prime}-f g^{\prime}}{g^2} $ Set $f=\sinh , g=\cosh$ to get $ \frac{d \tanh }{d x}=\frac{\cosh \cdot \sinh ^{\prime}-\sinh \cdot \cosh ^{\prime}}{\cosh ^2} $ Now, $ \begin{aligned} & \sinh ^{\prime}=\frac{1}{2}\left(e^x+e^{-x}\right)=\cosh \\ & \cosh ^{\prime}=\frac{1}{2}\left(e^x-e^{-x}\right)=\sinh \end{aligned} $ Thus, $ \frac{d \tanh }{d x}=\frac{\cosh ^2-\sinh ^2}{\cosh ^2}=1-\left(\frac{\sinh }{\cosh }\right)^2=1-\tanh ^2 $ --- ## References