Q1423: Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ with respect to...

Class CBSE Class 12 Mathematics Derivatives Q #1423

KNOWLEDGE BASED

UNDERSTAND

2 Marks 2025 AISSCE(Board Exam) VSA

Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ with respect to x.

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AI Tutor Explanation

Detailed Solution

Step 1: Define the function

Let $y = \frac{\sin x}{\sqrt{\cos x}}$. We need to find $\frac{dy}{dx}$.

Step 2: Apply the quotient rule

The quotient rule states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. Here, $u = \sin x$ and $v = \sqrt{\cos x} = (\cos x)^{1/2}$.

Step 3: Find the derivatives of u and v

We have $\frac{du}{dx} = \frac{d}{dx}(\sin x) = \cos x$. Also, $\frac{dv}{dx} = \frac{d}{dx}(\sqrt{\cos x}) = \frac{d}{dx}((\cos x)^{1/2}) = \frac{1}{2}(\cos x)^{-1/2} \cdot (-\sin x) = -\frac{\sin x}{2\sqrt{\cos x}}$.

Step 4: Apply the quotient rule formula

Now, we substitute these into the quotient rule formula: $$ \frac{dy}{dx} = \frac{\sqrt{\cos x} \cdot \cos x - \sin x \cdot \left(-\frac{\sin x}{2\sqrt{\cos x}}\right)}{(\sqrt{\cos x})^2} $$

Step 5: Simplify the expression

$$ \frac{dy}{dx} = \frac{\cos x \sqrt{\cos x} + \frac{\sin^2 x}{2\sqrt{\cos x}}}{\cos x} = \frac{\frac{2\cos^2 x + \sin^2 x}{2\sqrt{\cos x}}}{\cos x} = \frac{2\cos^2 x + \sin^2 x}{2\cos x \sqrt{\cos x}} $$ $$ \frac{dy}{dx} = \frac{2\cos^2 x + \sin^2 x}{2\cos^{3/2} x} = \frac{2\cos^2 x + (1 - \cos^2 x)}{2\cos^{3/2} x} = \frac{\cos^2 x + 1}{2\cos^{3/2} x} $$

Step 6: Final Answer

Therefore, $\frac{dy}{dx} = \frac{\cos^2 x + 1}{2\cos^{3/2} x}$.

Final Answer: $\frac{\cos^2 x + 1}{2\cos^{3/2} x}$

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because it requires the student to apply the quotient rule and chain rule to differentiate the given function. The student needs to understand the rules of differentiation and apply them correctly.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (differentiation using quotient rule and chain rule) to arrive at the solution.<\/span>

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, which is a core concept in the syllabus.