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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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