Q1444: Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sq...

Class CBSE Class 12 Mathematics Derivatives Q #1444

COMPETENCY BASED

APPLY

2 Marks 2025 AISSCE(Board Exam) VSA

Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.

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AI Tutor Explanation

Detailed Solution

Step 1: Define the functions

Let $u = \sqrt{e^{\sqrt{2x}}}$ and $v = e^{\sqrt{2x}}$. We need to find $\frac{du}{dv}$.

Step 2: Express u in terms of v

We have $u = \sqrt{v}$, since $u = \sqrt{e^{\sqrt{2x}}} = \sqrt{v}$.

Step 3: Differentiate u with respect to v

Now, we differentiate $u$ with respect to $v$: $$ \frac{du}{dv} = \frac{d}{dv}(\sqrt{v}) = \frac{d}{dv}(v^{\frac{1}{2}}) = \frac{1}{2}v^{\frac{1}{2} - 1} = \frac{1}{2}v^{-\frac{1}{2}} = \frac{1}{2\sqrt{v}} $$

Step 4: Substitute the value of v

Substitute $v = e^{\sqrt{2x}}$ back into the expression: $$ \frac{du}{dv} = \frac{1}{2\sqrt{e^{\sqrt{2x}}}} $$

Step 5: Simplify the expression

We can simplify the expression further: $$ \frac{du}{dv} = \frac{1}{2\sqrt{e^{\sqrt{2x}}}} = \frac{1}{2e^{\frac{\sqrt{2x}}{2}}} $$

Final Answer: $\frac{1}{2e^{\frac{\sqrt{2x}}{2}}}$

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply the chain rule and differentiation formulas to solve the problem.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply the rules of differentiation, specifically the chain rule, to find the derivative. This involves a series of steps and procedures.

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 differentiate with respect to another function rather than a variable.