Before You Begin

Question #41230

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a real-valued function satisfying the implicit equation $\arcsin(x^2 - y^2) = \arccos(2xy)$ . The value of $\frac{dy}{dx}$ at $x = \frac{1}{2}$ is:

Options

A.
$-\frac{1}{\sqrt{3}}$
B.
$\frac{1}{\sqrt{3}}$
C.
$-\sqrt{3}$
D.
\text{Does not exist}

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

Domain Constraint Intersections|Implicit Differentiation Implicit Domain|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the 'Implicit to Explicit Conversion' trap involving inverse trigonometric functions.

Students commonly square and add the arguments to obtain the algebraic curve $(x^2-y^2)^2 + (2xy)^2 = 1$ , which simplifies to the circle $x^2+y^2=1$ .

They then implicitly differentiate this simplified equation to find $dy/dx = -x/y$ , yielding $-1/\sqrt{3}$ at $x=1/2$ .

However, this ignores the implicit domain constraints imposed by the principal branches of the inverse trigonometric functions.

Equating $\arcsin(A) = \arccos(B)$ requires the common angle to be in $[0, \pi/2]$ , forcing $A \ge 0$ and $B \ge 0$ .

This restricts the valid domain to $|x| \ge 1/\sqrt{2}$ .

Since $x=1/2$ is outside this domain, the function is undefined there, making the derivative non-existent.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Domain Constraint Intersections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free