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

1
by leenaroy13

2015-03-23T22:40:22+05:30

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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:

Left side differential coefficient 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

So the answer is none of the given options.

option (b) log x / x is the derivative of Log (Log x).
option c: (x log x)^-1 is the derivative of Log (log x)
correction: option b: log x /x is the derivative of (Log x)^2 / 2.
hope it is more clear now.
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