If y=x logx+1+x2)-1+x2 then dydx=

y=x log x+1+x2-1+x2 dydx=xx+1+x2 1+121+x2(2x) + log x+1+x2-1x1+x2(2x) =x1+x2+log x+1+x2-x1+x2 =sin h-1 x

If y=x logx+1+x2)-1+x2 then dydx=

y=x log x+1+x2-1+x2 dydx=xx+1+x2 1+121+x2(2x) + log x+1+x2-1x1+x2(2x) =x1+x2+log x+1+x2-x1+x2 =sin h-1 x

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$If y = x \log |x + \sqrt{1 + x^{2})} - \sqrt{1 + x^{2}} then \frac{dy}{dx} =|$

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

$\frac{1}{2} \log (x + \sqrt{1 + x^{2}})$

$\sin h^{- 1} x$

$\cosh^{- 1} x$

answer is C.

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Detailed Solution

$y = x \log |x + \sqrt{1 + x^{2}}| - \sqrt{1 + x^{2}} \frac{dy}{dx} = \frac{x}{x + \sqrt{1 + x^{2}}} [1 + \frac{1}{2 \sqrt{1 + x^{2}}} (2 x)] + \log |x + \sqrt{1 + x^{2}}| - \frac{1}{x \sqrt{1 + x^{2}}} (2 x) = \frac{x}{\sqrt{1 + x^{2}}} + \log |x + \sqrt{1 + x^{2}}| - \frac{x}{\sqrt{1 + x^{2}}} = \sin h^{- 1} x$