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Differentiate the functions w.r.t. x. x^ x cos x + x ^ 2 + x / x ^2 - 1

 Q10  Differentiate the functions w.r.t. x.    x ^ {x \cos x} + \frac{x^2 + 1 }{x^2 -1 }

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Given function is
x ^ {x \cos x} + \frac{x^2 + 1 }{x^2 -1 }
Take t = x^{x\cos x}
Take log on both the sides
\log t =x\cos x \log x
Now, differentiate w.r.t. x 
we get,
\frac{1}{t}\frac{dt}{dx} = \cos x\log x-x\sin x\log x + \frac{1}{x}.x.\cos x\\ \frac{dt}{dx}= t.\left (\log x(\cos x-x\sin x)+ \cos x \right ) = x^{x\cos x}\left ( \log x(\cos x-x\sin x)+ \cos x \right )
Similarly, 
take   k = \frac{x^2+1}{x^2-1}
Now. differentiate it w.r.t. x
we get,
\frac{dk}{dx} = \frac{2x(x^2-1)-2x(x^2+1)}{(x^2-1)^2} = \frac{2x^3-2x-2x^3-2x}{(x^2-1)^2} = \frac{-4x}{(x^2-1)^2}
Now,
\frac{dy}{dx} = \frac{dt}{dx}+\frac{dk}{dx}
\frac{dy}{dx} = x^{x\cos x}\left ( \log x(\cos x-x\sin x)+ \cos x \right )-\frac{4x}{(x^2-1)^2}
Therefore, the answer is  x^{x\cos x}\left ( \cos x(\log x+1)-x\sin x\log x\right )-\frac{4x}{(x^2-1)^2}

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