Question

what is differentiation of???​

Answer 1

To Differentiate :

${ \cot}^{ - 1} ( \sqrt{1 + {x}^{2} } + x)$

Let

$x = \cot \alpha$

$\implies { \cot}^{ - 1} ( \sqrt{1 + { \cot}^{2} \alpha } + \cot( \alpha ) ) \\ \\ \implies { \cot}^{ - 1} ( \sqrt{ { \cosec}^{2} \alpha } + \cot \alpha ) \\ \\ \implies \: { \cot}^{ - 1} ( \frac{1}{ \sin( \alpha ) } + \frac{ \cos( \alpha ) }{ \sin( \alpha ) } ) \\ \\ \implies { \cot}^{ - 1} ( \frac{1 + \cos( \alpha ) }{ \sin( \alpha ) } ) \\ \\ \implies { \cot}^{ - 1} ( \frac{2 { \cos}^{2} \frac{ \alpha }{2} }{2 \sin \frac{ \alpha }{2} \cos \frac{ \alpha }{2} } ) \\ \\ \implies \: { \cot}^{ - 1} ( \frac{ \cos \frac{ \alpha }{2} }{ \sin \frac{ \alpha }{2} } ) \\ \\ \implies { \cot}^{ - 1} ( \cot \frac{ \alpha }{2}) \\ \\ \implies \: \frac{ \alpha }{2}$

NOW PUT THE VALUE OF ALPHA :

$\star \: \frac{1}{2} ( { \cot}^{ - 1} x)$

Differentiate w.r.t. X

$\star \: \frac{d}{dx} \: ( \frac{1}{2} ( { \cot}^{ - 1} x)) \\ \\ \star \: \frac{1}{2} ( - \frac{1}{1 + {x}^{2} } ) \\ \\ \star \: - \frac{1}{2(1 + {x}^{2} )}$

Formula :

$\star \: \boxed{ \green{ \frac{d}{dx} { \cot}^{ - 1} x = - \frac{1}{1 + {x}^{2} } }}\\ \\ \star \: \boxed{ \red{1 + \cos( \alpha ) = 2 { \cos}^{2} \frac{ \alpha }{2} }}$

Answer 2

Step-by-step explanation:

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