Inverse Trigonometric functions

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

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

$$ 3. \ \frac {d}{dx} \tan^{-1}x = \frac {1}{1+x^2} $$

$$ 4. \ \frac {d}{dx} \cot^{-1}x = \frac {-1}{1+x^2} $$

$$ 5. \ \frac {d}{dx} \sec^{-1}x = \frac {1}{x \sqrt {x^2-1}}, \ \vert x \vert \gt 1 $$

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

Example 1:

$$ Differentiate \ y = \frac {1}{ \sin^{-1}x} $$

Solution:

$$ \frac {dy}{dx} = \frac {d}{dx} \left( \sin^{-1}x \right)^{-1} $$

$$ \frac {dy}{dx} = - \left( \sin^{-1}x \right)^{-2} \frac {d}{dx} \left( \sin^{-1}x \right) $$

$$ \frac {dy}{dx} = - \frac {1}{ \left( \sin^{-1}x \right)^{2} \sqrt {1-x^2}} $$

Example 2:

$$ Differentiate \ f(x) = x \tan^{-1} \sqrt {x} $$

Solution:

$$ f(x) = \tan^{-1} \sqrt {x} + x \frac {1}{1+ \left( \sqrt {x} \right)^2 } \frac 12 x^{ - \frac 12} $$

$$ f(x) = \tan^{-1} \sqrt {x} + \frac { \sqrt {x}}{2(1+x)} $$