Question

differentiate the following
please help me to solve this question I want it urgent​

Answer 1

Step-by-step explanation:

We have,

$y = \sqrt{ \frac{ {x}^{2} + {a}^{2} }{ {x}^{2} - {a}^{2} } } . \sqrt[3]{ \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } }$

$\implies \: y = \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ - \frac{1}{2} } . \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg)^{ \frac{1}{3} }$

$\implies \: y = \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ \frac{1}{3} - \frac{1}{2} }$

$\implies \: y = \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ - \frac{1}{3} }$

$\implies \: y = \bigg( \frac{ {x}^{2} + {a}^{2} }{ {x}^{2} - {a}^{2} } \bigg) ^{ \frac{1}{3} }$

Differentiating both sides w.r.t. x,

$\implies \: \frac{dy}{dx} = \frac{1}{3} \bigg( \frac{ {x}^{2} + {a}^{2} }{ {x}^{2} - {a}^{2} } \bigg) ^{ \frac{1}{3} - 1 } . \frac{d}{dx} \bigg \{ \frac{ {x}^{2} + {a}^{2} }{ {x}^{2} - {a}^{2} } \bigg \}$

$\implies \: \frac{dy}{dx} = \frac{1}{3} \bigg( \frac{ {x}^{2} + {a}^{2} }{ {x}^{2} - {a}^{2} } \bigg) ^{ - \frac{2}{3} } . \bigg \{ \frac{( {x}^{2} - {a}^{2} ) .\frac{d}{dx} ({x}^{2} + {a}^{2} ) - ( {x}^{2} + {a}^{2}). \frac{d}{dx} ( {x}^{2} - a ^{2} ) }{ ({x}^{2} - {a}^{2} )^{2} } \bigg \}$

$\implies \: \frac{dy}{dx} = \frac{1}{3} \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ \frac{2}{3} } . \bigg \{ \frac{2x( {x}^{2} - {a}^{2} ) - 2x( {x}^{2} + {a}^{2}) }{ ({x}^{2} - {a}^{2} )^{2} } \bigg \}$

$\implies \: \frac{dy}{dx} = \frac{1}{3} \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ \frac{2}{3} } . \bigg \{ \frac{2 {x}^{3} - 2 {a}^{2} x - 2 {x}^{3} - 2{a}^{2}x}{ ({x}^{2} - {a}^{2} )^{2} } \bigg \}$

$\implies \: \frac{dy}{dx} = \frac{1}{3} \bigg( \frac{ {x}^{2} - {a}^{2} }{ {x}^{2} + {a}^{2} } \bigg) ^{ \frac{2}{3} } . \bigg \{ \frac{ - 4 {a}^{2} x }{ ({x}^{2} - {a}^{2} )^{2} } \bigg \}$