Class CBSE Class 12 Mathematics Derivatives Q #603
COMPETENCY BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:
(A) \((\frac{3}{2})^{x}\frac{log~2}{log~3}\)
(B) \((\frac{2}{3})^{x}\frac{log~3}{log~2}\)
(C) \((\frac{2}{3})^{x}\frac{log~2}{log~3}\)
(D) \((\frac{3}{2})^{x}\frac{log~3}{log~2}\)
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AI Tutor Explanation

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Step-by-Step Solution

Let \(u = 2^x\) and \(v = 3^x\). We want to find \(\frac{du}{dv}\).
First, find \(\frac{du}{dx}\) and \(\frac{dv}{dx}\): \(\frac{du}{dx} = \frac{d}{dx}(2^x) = 2^x \log 2\) \(\frac{dv}{dx} = \frac{d}{dx}(3^x) = 3^x \log 3\)
Now, use the chain rule: \(\frac{du}{dv} = \frac{du/dx}{dv/dx}\) \(\frac{du}{dv} = \frac{2^x \log 2}{3^x \log 3} = (\frac{2}{3})^x \frac{\log 2}{\log 3}\)

Correct Answer: \((\frac{2}{3})^{x}\frac{log~2}{log~3}\)

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 and derivative formulas to solve the problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to execute a sequence of steps involving differentiation and application of logarithmic properties.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply differentiation techniques in a non-standard way, requiring them to find the derivative of one function with respect to another, rather than a direct application of textbook formulas.

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