Find the derivative of $f(x)=7x\ln |x|$ Find the derivative of
$$f(x)=7x\ln|x|$$
How do they get the answer
$$f'(x)= 7\ln|x|+7$$

Since $\dfrac{d}{dx}\vert x\vert=\dfrac{\vert x\vert}{x}$
\begin{eqnarray}
\frac{d}{dx}7x\ln\vert x\vert &=&7\ln\vert x\vert+7x\cdot\frac{1}{\vert x\vert}\cdot\frac{\vert x\vert}{x}\\
&=&7\ln\vert x\vert+7
\end{eqnarray}
Proof of derivative of $\vert x\vert$:
\begin{eqnarray}
\dfrac{d}{dx}\vert x\vert&=&\lim_{h\to0}\frac{\vert x+h\vert-\vert x\vert}{h}\\
&=& \lim_{h\to0}\frac{\vert x+h\vert-\vert x\vert}{h}\cdot\frac{\vert x+h\vert+\vert x\vert}{\vert x+h\vert+\vert x\vert}\\
&=&\lim_{h\to0}\frac{\vert x+h\vert^2-\vert x\vert^2}{h\cdot\left(\vert x+h\vert+\vert x\vert\right)}\\
&=&\lim_{h\to0}\frac{2xh+h^2}{h\cdot\left(\vert x+h\vert+\vert x\vert\right)}\\
&=&\lim_{h\to0}\frac{2x+h}{\vert x+h\vert+\vert x\vert}\\
&=&\frac{2x}{2\vert x\vert}\\
&=&\frac{x}{\vert x\vert}=\frac{\vert x\vert}{x}
\end{eqnarray}
