Question

differentiate the following function by proper substitution​

Answer 1

$Given \: \: Question \: \: Is \: \: \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{t {}^{2} - 1} }{t {}^{2} } ) \\ \\Answer \: \: \: \\ \\ put \: \: \: t = sec(x) \\ \\ x = \sec {}^{ - 1} (t) \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{ \sec {}^{2} (x) - 1} }{sec {}^{2} (x)} ) \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{ \tan {}^{2} (x) } }{ \sec {}^{2} (x) } ) \\ \\ \sin {}^{ - 1} ( \frac{2 \tan(x) }{ \sec {}^{2} ( x ) } ) \\ \\ \sin {}^{ - 1} ( \frac{2 \frac{ \sin(x) }{ \cos(x) } }{ \frac{1}{ \cos {}^{2} (x) } } ) \\ \\ \sin {}^{-1} (2 \sin(x) \cos(x) ) \\ \\ \sin {}^{ - 1} ( \sin(2x) ) \\ \\ 2x \\ \\ 2 \sec {}^{ - 1} (t) \\ \\ f(x) = 2 \sec {}^{ - 1} ( t) \\ \\ Now \: Differentiate \: both \: sides \: with \: respect \: \\ to \: t \: we \: have. \\ \\ f {}^{1} (x) = \frac{2}{x \sqrt{x {}^{2} - 1} } \\ \\ so \: the \: Differentiation \: \: of \\ \sin( \frac{2 \sqrt{t {}^{2} - 1} }{t {}^{2} } ) \: \: \: \: \: \: is \: \: \: \: \frac{2}{x \sqrt{x {}^{2} - 1} } \\ \\ \\ Note \: \\ \\ 1) \: \: \: \: 1 + \tan {}^{2} (x) = \sec {}^{2} (x) \\ \\ 2) \: \: \: \: sin(2x) = 2 \sin(x) \cos(x) \\ \\ 3) \: \: \: Differentiation \: \: of \: \: \sec {}^{ - 1} (x) \: \: \: is \: \: \\ \\ \frac{1}{x \sqrt{x {}^{2} - 1} }$

Answer 2

$\begin{lgathered} Question \: \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{t {}^{2} - 1} }{t {}^{2} } ) \\ \\Answer \: \: \: \\ \\ Put \: \: \: t = sec(x) \\ \\ x = \sec {}^{ - 1} (t) \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{ \sec {}^{2} (x) - 1} }{sec {}^{2} (x)} ) \\ \\ \sin {}^{ - 1} ( \frac{2 \sqrt{ \tan {}^{2} (x) } }{ \sec {}^{2} (x) } ) \\ \\ \sin {}^{ - 1} ( \frac{2 \tan(x) }{ \sec {}^{2} ( x ) } ) \\ \\ \sin {}^{ - 1} ( \frac{2 \frac{ \sin(x) }{ \cos(x) } }{ \frac{1}{ \cos {}^{2} (x) } } ) \\ \\ \sin {}^{-1} (2 \sin(x) \cos(x) ) \\ \\ \sin {}^{ - 1} ( \sin(2x) ) \\ \\ 2x \\ \\ 2 \sec {}^{ - 1} (t) \\ \\ f(x) = 2 \sec {}^{ - 1} ( t) \\ \\ \\ Now \: Differentiate \: Both \: Sides \: With \: Respect \: \\ To \: t \: We \: Have. \\ \\ \\ f {}^{1} (x) = \frac{2}{x \sqrt{x {}^{2} - 1} } \\ \\ \\ So \: The \: Differentiation \: \: of \\ \sin( \frac{2 \sqrt{t {}^{2} - 1} }{t {}^{2} } ) \: \: \: \: \: \: is \: \: \: \: \frac{2}{x \sqrt{x {}^{2} - 1} } \\ \\ \\ \end{lgathered}$