If $y = {\cos ^{ - 1}}x,$ find $\cfrac{{{d^2}y}}{{d{x^2}}}$in terms of y alone.

$y = {\cos ^{ - 1}}x \Rightarrow$ $x = \cos y$ …(i)
Differentiating (i) w.r.t. x, we get
$1 = - \sin y\cfrac{{dy}}{{dx}}$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = - \cos ecy$ …(ii)

Differentiating (ii) w.r.t. x, we get

$\cfrac{{{d^2}y}}{{d{x^2}}} = \cos ecy\cot y\cfrac{{dy}}{{dx}} = - \cos e{c^2}y\cot y$