Derivative of $Tanh(x)$

Derivative of $tanh(x)$. It is used to gradient decent minimal of cost function.

$tanh(x) = {sinh(x) \over cosh(x)}$

Previous: Derivative of $sinh(x)$

$sinh(x) = {e^x-e^{-x} \over 2}$ $\frac d{dx}sinh(x) = \frac d{dx}\bigl({e^x-e^{-x} \over 2}\bigl)$ $= \frac 1 2 \frac d{dx}\bigl(e^x\bigl)-\frac 1 2\frac d{dx}\bigl(e^{-x}\bigl)$ $= \frac 1 2 e^x + \frac 1 2 e^{-x}$ $= {e^x+e^{-x} \over 2}$ $= cosh(x)$

Previous: Derivative of $cosh(x)$

$cosh(x) = {e^x+e^{-x} \over 2}$ $\frac d{dx}cosh(x) = \frac d{dx}\bigl({e^x+e^{-x} \over 2}\bigl)$ $= \frac 1 2 \frac d{dx}\bigl(e^x\bigl)+\frac 1 2\frac d{dx}\bigl(e^{-x}\bigl)$ $= \frac 1 2 e^x - \frac 1 2 e^{-x}$ $= {e^x-e^{-x} \over 2}$ $= sinh(x)$

Now we can derive $tanh(x)$…

$\frac d{dx}tanh(x) = \frac d{dx}\bigl({sinh(x) \over cosh(x)}\bigl)$ $= {cosh(x)\frac d{dx}sinh(x) - sinh(x)\frac d{dx}cosh(x) \over cosh^2(x)}$ $\frac d{dx}sinh(x) = cosh(x)$ $\frac d{dx}cosh(x) = sinh(x)$ $= {cosh(x)cosh(x) - sinh(x)sinh(x) \over cosh^2(x)}$ $=1-tanh^2(x)$