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Question #41232

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the curve defined implicitly by the equation $\arcsin\left( \frac{2xy}{x^2+y^2} \right) + \arctan\left( \frac{2xy}{x^2-y^2} \right) = \ln(x^2+y^2)$ Find the value of $\frac{dy}{dx}$ at the point $P\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ .

Options

A.
$-\frac{1}{\sqrt{3}}$
B.
$8 + 5\sqrt{3}$
C.
$\frac{1}{\sqrt{3}}$
D.
$-\sqrt{3}$

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

Algebraic-Trigonometric Graphical Mapping|Homogeneous Function Homogeneity|Implicit Differentiation Implicit Domain

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle implicit differentiation in the presence of inverse trigonometric functions with restricted domains.

The trap lies in the mechanical application of the chain rule or standard identities (like assuming $\arcsin(\frac{2v}{1+v^2}) + \arctan(\frac{2v}{1-v^2}) = 4\arctan v$ universally).

Because the point $P$ has $y > x$ (i.e., $v > 1$ ), the standard identity fails due to the implicit domain constraints of the ITFs.

Students who blindly differentiate without analyzing the local branch of the homogeneous function will miss the cancellation that reduces the LHS to 0, leading them to the incorrect derivative $8+5\sqrt{3}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Algebraic-Trigonometric Graphical Mapping
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free