Differentiate the following functions with respect to x: 2^x x

Question

Differentiate the following functions with respect to x:$\frac{2^\text{x}\cot\text{x}}{\sqrt{\text{x}}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{2^\text{x}\cot\text{x}}{\sqrt{\text{x}}}\Big)$
Using quotient rule, we get
$\frac{\sqrt{\text{x}}\frac{\text{d}}{\text{dx}}(2^\text{x}\cot\text{x})-(2^\text{x}\cot\text{x})\frac{\text{d}}{\text{dx}}(\sqrt{\text{x}})}{(\sqrt{\text{x}})^2}$
$=\frac{\sqrt{\text{x}}\Big(2^\text{x}\frac{\text{d}}{\text{dx}}\cot\text{x}+\cot\text{x}\frac{\text{d}}{\text{dx}}2^\text{x}\Big)-2^\text{x}\cot\text{x}\times\frac{1}{2}\text{x}^{-\frac{1}{2}}}{(\sqrt{\text{x}})^2}$
$=\frac{\sqrt{\text{x}}(2^\text{x}-\text{cosec}^2\text{x}+\cot\text{x}\times\log2\times2^\text{x})-2^\text{x}\cot\text{x}\times\frac{1}{2\sqrt{\text{x}}}}{(\sqrt{\text{x}})^2}$
$=\frac{2^\text{x}\Big\{-\text{x}\text{cosec}^2\text{x}+\text{x}\cot\text{x}\times\log2-\Big(\frac{1}{2}\Big)\cot\text{x}\Big\}}{(\sqrt{\text{x}})^2\times\sqrt{\text{x}}}$
$=\frac{2^\text{x}\Big(-\text{x}\text{cosec}^2\text{x}+\text{x}\cot\text{x}\times\log2-\Big(\frac{1}{2}\Big)\cot\text{x}\Big)}{\text{x}^{\frac{3}{2}}}$