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Question #66273

Difficulty: 8/10 | 🟢 Free

The Question

Let $u(x)$ be a differentiable function such that $u(x) \in \left( \frac{\pi}{2}, \frac{3\pi}{4} \right)$ for all $x \in \mathbb{R}$ . Define the function $f(x)$ as: $f(x) = \sec^{-1}\left( \frac{1 + \tan^2(u(x))}{1 - \tan^2(u(x))} \right) + \csc^{-1}\left( \frac{1 + \tan^2(u(x))}{2 \tan(u(x))} \right)$ Which of the following represents $f'(x)$ ?

Options

A.
$-4u'(x)$
B.
$4u'(x)$
C.
$0$
D.
$-2u'(x)$

Check Your Answer

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

Chain Rule Inner Function Differentiability|Tangent Half-Angle Substitution|The Quadrant Adjustment Logic

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to correctly apply the chain rule to composite inverse trigonometric functions while navigating branch cuts (Quadrant Adjustment Logic).

The trap lies in the blind application of the identity $f^{-1}(f(x)) = x$ .

Because the inner function $2u(x)$ falls in the third quadrant $(\pi, 3\pi/2)$ , it lies outside the principal ranges of both $\sec^{-1}$ and $\csc^{-1}$ .

Students who ignore this will incorrectly simplify the function to $4u(x)$ , leading to the trap answer $4u'(x)$ .

Correctly identifying the branch cuts $2\pi - 2u(x)$ and $\pi - 2u(x)$ is essential to finding the true derivative, highlighting a classic JEE Advanced filter mechanism.

Quick Facts About This Question

Question Information

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