Question

Differentiate the following w.r.t.x:
${\cot }^{-1}\left(\frac{1}{\sqrt{x}}\right)$

Answer 1

Let $y={\cot }^{-1}\left(\frac{1}{\sqrt{x}}\right)$

On differentiating w.r.t $x$, we get

$\frac{dy}{dx}=-\frac{1}{1+{\left(\frac{1}{\sqrt{x}}\right)}^2}\times \left(\frac{\sqrt{x}\times 0-1\times \frac{1}{2\sqrt{x}}}{(\sqrt{x}{)}^2}\right)$

$\frac{dy}{dx}=\frac{1}{1+\left(\frac{1}{x}\right)}\times \left(\frac{1}{2(x{)}^{\frac{3}{2}}}\right)$

$\frac{dy}{dx}=\frac{x}{x+1}\times \left(\frac{1}{2x(x{)}^{\frac{1}{2}}}\right)$

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

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

Hence, this is the answer.