derivative of inverse hyperbolic functions
\boxed{\frac d{dx}\left(\sin h^{-1}x\right)=\frac1{\sqrt{1+x^2}}} where x\in R
\boxed{\frac d{dx}\left(\cos h^{-1}x\right)=\frac1{\sqrt{x^2-1}}} where x>1
\boxed{\frac d{dx}\left(cosech^{-1}x\right)=\frac{-1}{{\displaystyle X}\sqrt{x^2+1}}} where x>0
\boxed{\frac d{dx}\left(sech^{-1}x\right)=\frac{-1}{{\displaystyle X}\sqrt{1-x^2}}} where 0<x<1
\boxed{\frac d{dx}\left(Tanh^{-1}x\right)=\frac1{1-x^2}} where \left|x\right|<1
\boxed{\frac d{dx}\left(coth^{-1}x\right)=\frac1{1-x^2}} where \left|x\right|>1
Derivatives of sin inverse hyperbolic function
Let.
y=\sin h^{-1}x x=\sin hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}\sin hy \cos hy.\frac{dy}{dx}=1 \frac{dy}{dx}=\frac1{\cos hy} \frac{dy}{dx}=\frac1{\sqrt{\cos{\displaystyle{\displaystyle h}^2}{\displaystyle y}}}Now by using formula.
\boxed{\sin h^2y-\cos h^2y=1}
Now by using eq(1)
\frac{dy}{dx}=\frac1{\sqrt{\displaystyle\sin h^2y+1}}\boxed{\frac{dy}{dx}=\frac1{\sqrt{\displaystyle x^2+1}}}
\boxed{\frac d{dx}sinh^{-1}x=\frac1{\sqrt{\displaystyle x^2+1}}}
Derivatives of cos inverse hyperbolic function
Let.
y=\cos h^{-1}x x=\cos hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}\cos hy \sin hy.\frac{dy}{dx}=1 \frac{dy}{dx}=\frac1{\sin hy} \frac{dy}{dx}=\frac1{\sqrt{\sin{\displaystyle{\displaystyle h}^2}{\displaystyle y}}}Now by using formula.
\boxed{\sin h^2y-\cos h^2y=1}
Now by using eq(1)
\frac{dy}{dx}=\frac1{\sqrt{\displaystyle\cos h^2y-1}}\boxed{\frac{dy}{dx}=\frac1{\sqrt{\displaystyle x^2-1}}}
\boxed{\frac d{dx}cosh^{-1}x=\frac1{\sqrt{\displaystyle x^2-1}}}
Derivatives of Tan inverse hyperbolic function
Let.
y=Tan h^{-1}x x=Tan hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}Tan hy \sec h^2y.\frac{dy}{dx}=1 \frac{dy}{dx}=\frac1{sech^2 y} \frac{dy}{dx}=\frac1{{sec{\displaystyle{\displaystyle h}^2}{\displaystyle y}}}Now by using formula.
\boxed{Sec h^2y-Tan h^2y=1}
Now by using eq(1)
\frac{dy}{dx}=\frac1{{1-Tan h^2y}}\boxed{\frac{dy}{dx}=\frac1{{1- x^2}}}
\boxed{\frac d{dx}Tanh^{-1}=\frac1{{1- x^2}}}
Derivatives of cosec inverse hyperbolic function
Let.
y=\cosec h^{-1}x x=\cosec hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}\cosec hy -cosec hy.cot hy.\frac{dy}{dx}=1 \frac{dy}{dx}=-\frac1{cosec hy.cot hy}Now by using formula.
\boxed{\cosec h^2y=cot h^2y-1}
\frac{dy}{dx}=-\frac1{cosec hy\sqrt{\cosec{\displaystyle{\displaystyle h}^2}{\displaystyle y+1}}}Now by using eq(1)
\boxed{\frac{dy}{dx}=-\frac1{x\sqrt{x^2+1}}}\boxed{\frac d{dx}cosech^{-1}=-\frac1{x\sqrt{x^2+1}}}
Derivatives of sec inverse hyperbolic function
Let.
y=sec h^{-1}x x=sec hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}sec hy -sec hy.tan hy.\frac{dy}{dx}=1 \frac{dy}{dx}=-\frac1{sec hy.tan hy}Now by using formula.
\boxed{\sec h^2y=1-tan h^2y}
\frac{dy}{dx}=-\frac1{sec hy\sqrt{1-sec{\displaystyle{\displaystyle h}^2}{\displaystyle y}}}Now by using eq(1)
\boxed{frac{dy}{dx}=-\frac1{x\sqrt{1-x^2}}}
\boxed{\frac d{dx}sech^-{1}=-\frac1{x\sqrt{1-x^2}}}
Derivatives of cot inverse hyperbolic function
Let.
y=cot h^{-1}x x=cot hy.....(1)Differentiating w.r.t x
\frac d{dx}x=\frac d{dx}cot hy -cosec h^2y.\frac{dy}{dx}=1 \frac{dy}{dx}=\frac1{-cosech^2 y} \frac{dy}{dx}=-\frac1{{cosec{\displaystyle{\displaystyle h}^2}{\displaystyle y}}}Now by using formula.
\boxed{cosec h^2y=cot h^2y-1}
Now by using eq(1)
\frac{dy}{dx}=-\frac1{{cot h^2y-1}}\boxed{\frac{dy}{dx}=-\frac1{{1-x^2}}}
\boxed{\frac d{dx}coth^{-1}x=-\frac1{{1-x^2}}}
These are the required derivative of inverse hyperbolic functions we should know.