The derivative of  ${\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^3$ is

Let, ${\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^3$

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

$\begin{array}{c}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}{\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^3\\ =\frac{\mathrm{d}}{\mathrm{dx}}\left\{{\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}+3\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)\right\}\end{array}$

       $\begin{array}{c}=\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^3+\frac{1}{{\mathrm{x}}^3}+3\mathrm{x}+\frac{3}{\mathrm{x}}\right)\\ =3{\mathrm{x}}^2-\frac{3}{{\mathrm{x}}^4}+3-\frac{3}{{\mathrm{x}}^2}\end{array}$