Class CBSE Class 12 Mathematics Derivatives Q #1355
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2025 AISSCE(Board Exam) VSA
Differentiate $2^{\cos^{2}x}$ w.r.t $\cos^{2}x$.
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Detailed Solution

Step 1: Define the function

Let $y = 2^{\cos^2 x}$. We need to find $\frac{dy}{d(\cos^2 x)}$.

Step 2: Apply the chain rule

Let $u = \cos^2 x$. Then $y = 2^u$. We want to find $\frac{dy}{du}$.

Step 3: Differentiate with respect to u

We know that the derivative of $a^x$ with respect to $x$ is $a^x \ln a$. Therefore, the derivative of $2^u$ with respect to $u$ is $2^u \ln 2$.

Step 4: Substitute back for u

Substituting $u = \cos^2 x$ back into the expression, we get $\frac{dy}{du} = 2^{\cos^2 x} \ln 2$.

Final Answer: $2^{\cos^2 x} \ln 2$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires the student to comprehend the concept of differentiation and apply the chain rule.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the chain rule and differentiation of exponential functions, which are conceptual knowledge.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly tests a standard differentiation technique covered in the textbook.

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