Inverse Hyperbolic functions

$$ 1. \ \frac {d}{dx} \sin h^{-1}x = \frac {1}{\sqrt {1+x^2}} $$

$$ 2. \ \frac {d}{dx} \cos h^{-1}x = \frac {1}{\sqrt {x^2 -1}} $$

$$ 3. \ \frac {d}{dx} \tan h^{-1}x = \frac {1}{1-x^2}, \ \vert x \vert \lt 1 $$

$$ 4. \ \frac {d}{dx} \cot h^{-1}x = \frac {1}{x^2-1}, \ \vert x \vert \gt 1 $$

$$ 5. \ \frac {d}{dx} \sec h^{-1}x = \frac {-1}{x \sqrt {1-x^2}}, \ 0 \lt x \lt 1 $$

$$ 6. \ \frac {d}{dx} cosec \ h^{-1}x = \frac {-1}{x \sqrt {1+x^2}}, \ x \gt 0 $$

Example 1:

$$ Differentiate \ y= \tan h^{-1} (\sin x) $$

Solution:

$$ y'= \frac {1}{1- \sin^2x} \cdot \frac {d}{dx} ( \sin x) $$

$$ y'= \frac {1}{1- \sin^2x} \cdot \cos x $$

$$ y'= \frac { \cos x}{\cos^2x} $$

$$ y'= \sec x $$