Differentiability of $\log|x|$ It is known that
$$\dfrac{d}{dx}\log f(x) = \dfrac{1}{f(x)} f'(x)$$
but how does it is not applicable for $\log |x|$ as $|x|$ is not differentiable?

Since $\log(|x|)$ is not defined at $x=0$, we may investigate the differentiability for $x\not=0$. We  distinguish two cases depending on the sign of $x$.
For $x>0$, we know that $|x|=x$ and
$$\frac{d}{dx}\left(\log(|x|)\right)=\frac{d}{dx}\left(\log(x)\right)=\frac{1}{x}.$$
On the other hand, for $x<0$, we have that $|x|=-x$ and therefore
$$\frac{d}{dx}\left(\log(|x|)\right)=\frac{d}{dx}\left(\log(-x)\right)=\frac{1}{(-x)}\cdot (-1)=\frac{1}{x}.$$
So we may conclude that $\log(|x|)$ is differentiable for $x\not=0$, and its derivative is
$$\frac{d}{dx}\left(\log(|x|)\right)=\frac{1}{x}.$$
