Question

Find the derivative of $ \log \left| x \right| $ . ( $ x \ne 0 $ )
A. $ 1/x $
B. $ -1/x $
C. $ x $
D. $ -x $

Answer 1

Hint: It is mentioned in the question that x is not equal to 0 which means either x is greater than zero or less than zero. So, one time we will take x more than 0 and other times less than 0 and will find the values. We will solve the question accordingly and check both the answers which will come.

Complete step-by-step answer:
According to the question:
Given: x is not equal to 0 so either x is more than 0 or less than 0.
 $ x > 0 $ or \[x < 0\]
When x is more than zero ( $ x > 0 $ ) then value of x will be positive and
we will take $ \log \left( x \right) $
When x is less than zero ( $ x < 0 $ ) then value of x will be negative and
we will take $ \log ( - x) $
First we will take $ x > 0 $ ,
Let $ y = \log x $
Now we will differentiate this function with respect to x,
 $ \dfrac{{dy}}{{dx}} = \dfrac{1}{x}.1 $
 $ \dfrac{{dy}}{{dx}} = \dfrac{1}{x} $
Now take $ x < 0 $
Let $ y = \log \left( { - x} \right) $
Now we will differentiate this function with respect to x,
 $ \dfrac{{dy}}{{dx}} = \dfrac{1}{{ - x}}.\left( { - 1} \right) $
 $ \dfrac{{dy}}{{dx}} = \dfrac{1}{x} $
So, either we take x more than zero or less than zero, our answer is the same , which is $ \dfrac{1}{x} $ .
Hence, value of derivative of $ \log \left| x \right| $ is $ \dfrac{1}{x} $ . So option (1) is the correct answer.
So, the correct answer is “Option 1”.

Note: Whenever it is mentioned in the question that the value of the derivative is not equal to some digit then, we must consider all other digits to find the answer. In the above solution, differentiation is done with respect to x and by using the chain rule of differentiation.