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Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsDerivatives3 Marks
Question
Differentiate the following function with respect to $(\text{x})$:
$\Big(\text{x}+\frac{1}{\text{x}}\Big)\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)$
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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\text{x}+\frac{1}{\text{x}}\Big)\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)$
$=\Big(\text{x}+\frac{1}{\text{x}}\Big)\frac{\text{d}}{\text{dx}}\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)+\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)\frac{\text{d}}{\text{dx}}\Big(\text{x}+\frac{1}{\text{x}}\Big)$ [Using product rule]
$=\Big(\text{x}+\frac{1}{\text{x}}\Big)\Bigg(\frac{1}{2\sqrt{\text{x}}}-\frac{1}{2\text{x}^{\frac{3}{2}}}\Bigg)+\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)\Big(1-\frac{1}{\text{x}^2}\Big)$
$=\Bigg(\frac{\text{x}}{2\sqrt{\text{x}}}-\frac{\text{x}}{\text{2x}^{\frac{3}{2}}}+\frac{1}{\text{2x}^{\frac{3}{2}}}-\frac{1}{\text{2x}^{\frac{5}{2}}}\Bigg)+\Bigg(\sqrt{\text{x}}-\frac{\sqrt{\text{x}}}{\text{x}^2}+\frac{1}{\sqrt{\text{x}}}-\frac{1}{\text{x}^{\frac{5}{2}}}\Bigg)$
$=\Bigg(\frac{1}{2}\sqrt{\text{x}}-\frac{1}{2\sqrt{\text{x}}}+\frac{1}{\text{2x}^{\frac{3}{2}}}-\frac{1}{\text{2x}^{\frac{5}{2}}}+\sqrt{\text{x}}-\frac{1}{\text{x}^{\frac{3}{2}}}+\frac{1}{\sqrt{\text{x}}}-\frac{1}{\text{x}^{\frac{5}{2}}}\Bigg)$
$=\Bigg(\frac{3}{2}\sqrt{\text{x}}+\frac{1}{2}\sqrt{\text{x}}-\frac{1}{\text{2x}^{\frac{3}{2}}}-\frac{3}{\text{2x}^{\frac{5}{2}}}\Bigg)$
$=\frac{3}{2}\text{x}^{\frac{1}{2}}+\frac{1}{2}\text{x}^{\frac{-1}{2}}-\frac{1}{2}\text{x}^{\frac{-3}{2}}-\frac{3}{2}\text{x}^{\frac{-5}{2}}$

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