Question

Differentiate the following functions with respect to x :

$(x+\frac{1}{x})(\sqrt{x}+\frac{1}{\sqrt{x}})$

Answer 1

We have

$\frac{d}{dx}(x+\frac{1}{x})(\sqrt{x}+\frac{1}{\sqrt{x}})$

$=(x+\frac{1}{x})\frac{d}{dx}(\sqrt{x}+\frac{1}{\sqrt{x}})+(\sqrt{x}+\frac{1}{\sqrt{x}})\frac{d}{dx}(x+\frac{1}{x})\left[\text{Using product rule}\right]$

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

$=(\frac{x}{2\sqrt{x}}-\frac{x}{2x^{\frac{3}{2}}}+\frac{1}{2x^{\frac{3}{2}}}-\frac{1}{2x^{\frac{5}{2}}})+(\sqrt{x}-\frac{\sqrt{x}}{x^2}+\frac{1}{\sqrt{x}}-\frac{1}{x^{\frac{5}{2}}})$

$=(\frac{3}{2}+\frac{1}{2}\sqrt{x}-\frac{1}{2x^{\frac{3}{2}}}-\frac{3}{2x^{\frac{5}{2}}})=\frac{3}{2}{x}^{\frac{1}{2}}+\frac{1}{2}{x}^{\frac{-1}{2}}-\frac{1}{2}{x}^{\frac{-3}{2}}-\frac{3}{2}{x}^{\frac{-5}{2}}$