CBSE Class 12 Mathematics Derivatives Q #605
KNOWLEDGE BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
The derivative of \(\sin(x^{2})\) w.r.t. x, at \(x=\sqrt{\pi}\) is :
(A) 1
(B) -1
(C) \(-2\sqrt{\pi}\)
(D) \(2\sqrt{\pi}\)
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Step-by-Step Solution

  1. Let \(y = \sin(x^2)\).

  2. Differentiate \(y\) with respect to \(x\) using the chain rule:

    \(\frac{dy}{dx} = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x = 2x \cos(x^2)\)

  3. Substitute \(x = \sqrt{\pi}\) into the derivative:

    \(\frac{dy}{dx}\Big|_{x=\sqrt{\pi}} = 2\sqrt{\pi} \cos((\sqrt{\pi})^2) = 2\sqrt{\pi} \cos(\pi)\)

  4. Since \(\cos(\pi) = -1\), we have:

    \(\frac{dy}{dx}\Big|_{x=\sqrt{\pi}} = 2\sqrt{\pi} \cdot (-1) = -2\sqrt{\pi}\)

Correct Answer: -2√π

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the chain rule of differentiation and then substitute a given value to find the answer.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure (chain rule of differentiation) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of differentiation rules and their application.

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