Originally Posted by
galactus You could perhaps try it this way:
$\displaystyle \frac{d}{dx}[csc^{-1}(x)]$
$\displaystyle y=csc^{-1}(x), \;\ x=csc(y)$
Differentiate:
$\displaystyle \frac{d}{dx}{x}=\frac{d}{dx}[csc(y)]\Rightarrow{1=-csc(y)cot(y)}\frac{dy}{dx}$
$\displaystyle \frac{dy}{dx}=\frac{1}{-csc(y)cot(y)}$
$\displaystyle \frac{dy}{dx}=\frac{1}{-csc(csc^{-1}(x))cot(csc^{-1}(x))}$
$\displaystyle \frac{dy}{dx}=\frac{-1}{x\sqrt{x^{2}-1}}$
$\displaystyle \frac{d}{dx}[csc^{-1}(x)^{2}]$
Chain rule:
$\displaystyle 2csc^{-1}(x)\frac{d}{dx}[csc^{-1}(x)]$
$\displaystyle \boxed{\frac{-2csc^{-1}(x)}{x\sqrt{x^{2}-1}}}$