Question

if f(x)=|x| then what is the differential coefficient of f(logx) with respect to x?

Answer 1

f(x) = | x |
       =  x  ,  for x ≥ 0
       = - x ,  for x ≤ 0

f( Log x) =  | Log  x |
         = Log x ,    for  1 < x ≤ ∞
         = 0  for x = 1
         = - Log x    for  0 < x < 1
         = undefined  for  x ≤ 0

The function f (Log x) exists in (0, ∞).  It is continuous at all points.  Let us find the derivative from left side and right side of x = 1.

Right side Differential coefficient for x > 1:

$\frac{d}{dx}f(Log\ x)=\frac{d}{dx}Log\ x=\frac{1}{x},\ \ for\ x>1$

Left side differential coefficient for 0< x < 1 :

$\frac{d}{dx}(-log\ x)=-\frac{1}{x},\ \ for\ \ 0 < x < 1$

At x = 1,  the derivative from right side is 1 and from left side is -1.  So there is no derivative defined for x = 1. Otherwise, it is defined as:

Differential coefficient of f (Log x) : 
       1/x  for  x > 1
       undefined for x = 1
       -1/x  for  0 < x < 1
       undefined for x <= 0

So the answer is :

           it is     1/x * |x-1|/(x-1)  defined  for  x > 0

$\frac{1}{x}*\frac{x-1}{|x-1|},\ \ \ x>0$

So the answer is none of the given options.