Question

The derivative of $\log \left|x\right|$ is

• A. $\frac{1}{x},x>0$
• B. $\frac{1}{\left|x\right|},x\ne 0$
• C. $\frac{1}{x},x\ne 0$
• D. None of these

Answer 1

The correct option is C

$\frac{1}{x},x\ne 0$

Explanation for the correct option:

Finding derivative of $\log \left|x\right|$ when $x>0$

When, $x>0$ the logarithm function will be $y=\log x$

Differentiating the function $y$ with respect to $x$

$\frac{dy}{dx}=\frac{1}{x}$

So, derivative of $\log \left|x\right|$ when $x>0$ is $\frac{1}{x}$

when $x<0$ logarithm function will be $y=\log \left(-x\right)$

Differentiating the function $y$ with respect to $x$

$$\begin{gathered} \frac{dy}{dx}=\frac{1}{-x}(-1) \\ \frac{dy}{dx}=\frac{1}{x} \end{gathered}$$

So, derivative of $\log \left|x\right|$ when $x<0$ is $\frac{1}{x}$

For $x=0$ , $\log \left|x\right|$ is not defined hence, it's derivative don't exist.

Hence, derivative of $\log \left|x\right|$ for $x\ne 0$ is $\frac{1}{x}$

Therefore, the correct answer is option (C).