# Differentiate the functions given in w.r.t. $x :$ $x^{\large x \cos x} + \large\frac{x^2 + 1}{x^2 - 1}$

Toolbox:
• $\big(\large\frac{u}{v}\big)'=\large\frac{u'v-uv'}{v^2}$
• $\log m^{\large n}=n\log m$
Step 1:
Let $y=x^{\large\cos x}+\large\frac{x^2+1}{x^2-1}$
It is of the form $u+v$
Now $u=x^{\large x\cos x}$
Taking $\log$ on both sides
$\log u=\log x^{\large x\cos x}$
We know that $\log m^{\large n}=n\log m$
$\log u=x\cos x\log x$
Step 2:
Differentiating with respect to $x$
$\large\frac{1}{u}\frac{du}{dx}=$$1.\cos x\log x+(-\sin x)(x\log x)+\large\frac{1}{x}$$(x\cos x)$
$[(uvw)'=u'vw+v'uw]$
$\Rightarrow \cos x\log x-x\sin x\log x+\cos x$