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Question #41316

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x) > 0$ be a real-valued function defined on its maximal domain $D \subset \mathbb{R}$ by the implicit relation $y^{x \ln x} = e^{x^2 - x}$ . Let $L = \lim_{x \to 1} f'(x)$ . Which of the following represents the ordered pair $(L, f'(1))$ ?

Options

A.
$\left(\frac{e}{2}, \frac{e}{2}\right)$
B.
$\left(\frac{e}{2}, \text{Does not exist}\right)$
C.
$\left(e, e\right)$
D.
$\left(e, \text{Does not exist}\right)$

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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

Logarithmic Differentiation Domain|Implicit to Explicit Conversion|Domain Restrictions|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the conversion of an implicit equation to an explicit form via logarithmic differentiation, coupled with a lethal domain trap.

Students frequently misuse L'Hôpital's rule by calculating the limit of the derivative at a boundary point (here, $x=1$ ) and falsely concluding that this limit equals the derivative at that point.

However, the original implicit equation $y^{x \ln x} = e^{x^2 - x}$ degenerates to $1=1$ at $x=1$ , meaning the function is not uniquely defined there.

Consequently, while the limit of the derivative exists and evaluates to $e/2$ via L'Hôpital's rule, the derivative itself does not exist because $x=1$ is outside the function's maximal domain.

Quick Facts About This Question

Question Information

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