Answer :
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:
[tex]\frac{d}{dx}f(Log\ x)=\frac{d}{dx}Log\ x=\frac{1}{x},\ \ for\ x>1[/tex]
Left side differential coefficient for 0< x < 1 :
[tex]\frac{d}{dx}(-log\ x)=-\frac{1}{x},\ \ for\ \ 0 < x < 1[/tex]
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
[tex]\frac{1}{x}*\frac{x-1}{|x-1|},\ \ \ x>0[/tex]
So the answer is none of the given options.
= 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:
[tex]\frac{d}{dx}f(Log\ x)=\frac{d}{dx}Log\ x=\frac{1}{x},\ \ for\ x>1[/tex]
Left side differential coefficient for 0< x < 1 :
[tex]\frac{d}{dx}(-log\ x)=-\frac{1}{x},\ \ for\ \ 0 < x < 1[/tex]
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
[tex]\frac{1}{x}*\frac{x-1}{|x-1|},\ \ \ x>0[/tex]
So the answer is none of the given options.