Differentiate the following functions with respect to x: e^x- -x^… - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{e}^\text{x}-\tan\text{x}}{\cot\text{x}-\text{x}^\text{n}}$

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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{e}^\text{x}-\tan\text{x}}{\cot\text{x}-\text{x}^\text{n}}\Big)$$=\frac{(\cot\text{x}-\text{x}^\text{n})\frac{\text{d}}{\text{dx}}(\text{e}^\text{x}-\tan\text{x})-(\text{e}^\text{x}-\tan\text{x})\frac{\text{d}}{\text{dx}}(\cot\text{x}-\text{x}^\text{n})}{(\cot\text{x}-\text{x}^\text{n})^2}$
$=\frac{(\cot\text{x}-\text{x}^\text{n})(\text{e}^\text{x}-\sec^2\text{x})-(\text{e}^\text{x}-\tan\text{x})(-\text{cosec}^2\text{x}-\text{nx}^{\text{n}-1})}{(\cot\text{x}-\text{x}^\text{n})^2}$
$=\frac{(\cot\text{x}-\text{x}^\text{n})(\text{e}^\text{x}-\sec^2\text{x})+(\text{e}^\text{x}-\tan\text{x})(\text{cosec}^2\text{x}-\text{nx}^{\text{n}-1})}{(\cot\text{x}-\text{x}^\text{n})^2}$

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