Question

If y = log$\left|x\right|,x\ne 0,\mathrm{then}\frac{dy}{dx}$ = ___________________.

Answer 1

For $y=\log \left|x\right|$ to be defined,

$\left|x\right|\ne 0$

$\Rightarrow x\ne 0$

Now,

$y=\log \left|x\right|=\left\{\begin{array}{ll}\log x,& x>0\\ \log \left(-x\right),& x<0\end{array}\right.$

$\therefore \frac{dy}{dx}=\left\{\begin{array}{ll}\frac{1}{x},& x>0\\ \frac{1}{\left(-x\right)}\times \left(-1\right),& x<0\end{array}\right.$

$\Rightarrow \frac{dy}{dx}=\left\{\begin{array}{ll}\frac{1}{x},& x>0\\ \frac{1}{x},& x<0\end{array}\right.$

$\therefore \frac{dy}{dx}=\frac{1}{x},x\ne 0$


If y = log$\left|x\right|,x\ne 0,\mathrm{then}\frac{dy}{dx}$ = $\underline{\frac{1}{x}}$.