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Question #41308

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \ln\left(\frac{2x}{1-x^2}\right)$ and $g(x) = \tan x$ . Consider the composite function $h(x) = f(g(x))$ . If $S$ is the largest subset of $(0, \pi)$ on which the derivative $h'(x)$ exists and is continuous, then $S$ is equal to:

Options

A.
$(0, \frac{\pi}{4}) \cup (\frac{\pi}{4}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3\pi}{4}) \cup (\frac{3\pi}{4}, \pi)$
B.
$(0, \frac{\pi}{4}) \cup (\frac{\pi}{2}, \frac{3\pi}{4})$
C.
$(0, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \pi)$
D.
$(\frac{\pi}{4}, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, \pi)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Composition Continuity|Substitution (Change of Variable)|Logarithmic Differentiation Domain

Why This Question Is Tricky

Common Mistakes

This question targets the 'Domain Shrinkage' and 'Logarithmic Differentiation' traps.

Students frequently apply the chain rule to $f(g(x))$ and simplify the resulting derivative algebraically (here, obtaining $4\csc 4x$ ).

They then incorrectly determine the continuity of the derivative based solely on this final algebraic expression, ignoring the fundamental calculus principle that a derivative cannot exist where the original function is undefined.

The substitution $g(x) = \tan x$ into the logarithmic function restricts the domain, creating a lethal collision between the apparent domain of the simplified derivative and the actual domain of the composition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Composition Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free