Differentiate the following functions. x^ 5 - - Vidyadip

Question

Differentiate the following functions.
$\frac{\text{x}^{5}-\cos\text{x}}{\sin\text{x}}$

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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}^{5}-\cos\text{x}}{\sin\text{x}}\Big)$
$=\frac{\sin\text{x}\frac{\text{d}}{\text{dx}}-(\text{x}^{5}-\cos\text{x}).\frac{\text{d}}{\text{dx}}(\sin\text{x})}{\sin^{2}\text{x}}$
$=\frac{\sin\text{x}(5\text{x}^{4}+\sin\text{x})-(\text{x}^{5}-\cos\text{x})(\cos\text{x})}{\sin^{2}\text{x}}$
$=\frac{5\text{x}^{4}.\sin\text{x}+\sin^{2}\text{x}-\text{x}^{5}\cos\text{x}+\cos^{2}\text{x}}{\sin^{2}\text{x}}$
$=\frac{5\text{x}^{4}.\sin\text{x}-\text{x}^{5}\cos\text{x}+(\sin^{2}\text{x}+\cos^{2}\text{x})}{\sin^{2}\text{x}}$
$=\frac{5\text{x}^{4}\sin\text{x}-\text{x}^{5}\cos\text{x}+1}{\sin^{2}\text{x}}$
Hence, the required answer is $\frac{5\text{x}^{4}\sin\text{x}-\text{x}^{5}\cos\text{x}+1}{\sin^{2}\text{x}}.$

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