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[tex]DERIVATIVE \: \: OF \: \: \\ \: \: \: \: \:PARAMETRIC \: \: FUNCTIONS\\ \\ \\ x = a( \beta - \sin( \beta ) ) \\ y = a(1 - \cos( \beta ) \\ \\ \frac{dy}{d \beta } = a \sin( \beta ) \\ \\ \frac{dx}{d \beta } = a(1 - \cos( \beta ) \\ \\ \frac{dy}{dx} = \frac{a \sin( \beta ) }{a(1 - \cos( \beta ) } = \cot( \frac{ \beta }{2} ) \\ \\ \\ \frac{d^{2}y }{d {x}^{2} } = \frac{d}{dx} ( \ \ \cot ( \frac{ \beta }{2} ) ) = ( - \frac{1}{2} cosec ^{2} \frac{ \beta }{2} ) \times \frac{d \beta }{dx} \\ \\ \\ = (- \frac{1}{2} cosec^{2} \frac{ \beta }{2} ) \times \frac{1}{a(1 - \cos( \beta ) } \\ \\ \\ At \: \beta = \pi \\ \\ = ( - \frac{ 1}{2} {cosec}^{2}\frac{\pi}{2} ) \times \frac{1}{a(1 - \cos(\pi)) } \\ \\ = - \frac{1}{2} . {1}^{2} \times \frac{1}{a(1 - ( - 1))} \\ \\ \\ = - \frac{1}{4a} \: \: \: \: \: \: \: \: \: Ans. [/tex]

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