Question

If $y={cos}^{-1}x$, find $\frac{d^{2}y}{{dx}^2}$ in terms of 'y' only.

Answer 1

$\Rightarrow y=co{s}^{-1}x$

$\Rightarrow x=cosy$

Differentiating with respect to $x$

$1=-siny\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{-1}{siny}=-cosecy$

Again Diff.,$\frac{d^{2}y}{d{x}^2}=+coseycoty\frac{dy}{dx}$

$=cosecy.coty(-cosecy)$

$=\frac{-cosy}{si{n}^{3}y}$