Differentiate the following function with respect to (x): ( x +1 … - Vidyadip

Question

Differentiate the following function with respect to $(\text{x})$:$\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)^3$

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Answer

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

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