Question

Find the differentiation of w.r.t $x$
$si{n}^{-1}\sqrt{x}$

Answer 1

$y={\sin }^{-1}\sqrt{x}$

we need $\frac{dy}{dx}$

$\sin y=\sqrt{x}\Rightarrow \cos y=\sqrt{1-{\sin }^{2}y}=\sqrt{1-x}$

On differentiating

$\cos y\frac{dy}{dx}=\frac{1}{2\sqrt{x}}$

$\frac{dy}{dx}=\frac{\cos y}{2\sqrt{x}}=\frac{\sqrt{1-x}}{2\sqrt{x}}=\frac{1}{2}\sqrt{\frac{1-x}{x}}$

$\frac{dy}{dx}=\frac{1}{2}\sqrt{\frac{1}{x}-1}$

$\therefore$ differentiation of ${\sin }^{-1}\sqrt{x}$ is $\frac{1}{2}\sqrt{\frac{1}{x}-1}$