Question

Differentiate the following functions w.r.t.$x$.
$\sqrt{x+\frac{1}{x}}$.

Answer 1

Given $y=\sqrt{x+\frac{1}{x}}$

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

$=\frac{1}{2}\frac{\sqrt{x}}{\sqrt{x^2+1}}\frac{(x^2-1)}{x^2}$

$=\frac{1}{2}\frac{(x^2-1)}{\sqrt{x^2+1}.x^{2-\frac{1}{2}}}=\frac{1}{2}\frac{(x^2-1)}{\left(\sqrt{x^2+1}\right)\left(x^{\frac{3}{2}}\right)}$