Q1476: Differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-...

Class CBSE Class 12 Mathematics Derivatives Q #1476

KNOWLEDGE BASED

REMEMBER

3 Marks 2025 AISSCE(Board Exam) SA

Differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}$ with respect to x.

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Detailed Solution

Step 1: Apply the chain rule

We need to differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}$ with respect to $x$. We will use the chain rule repeatedly.

Step 2: Differentiate the outermost function

The outermost function is the square root. So, we have: $$\frac{dy}{dx} = \frac{1}{2\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}} \cdot \frac{d}{dx}\left[\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}\right]$$

Step 3: Differentiate the logarithm

Now, we differentiate the logarithm: $$\frac{d}{dx}\left[\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}\right] = \frac{1}{\sin\left(\frac{x^{3}}{3}-1\right)} \cdot \frac{d}{dx}\left[\sin\left(\frac{x^{3}}{3}-1\right)\right]$$

Step 4: Differentiate the sine function

Next, we differentiate the sine function: $$\frac{d}{dx}\left[\sin\left(\frac{x^{3}}{3}-1\right)\right] = \cos\left(\frac{x^{3}}{3}-1\right) \cdot \frac{d}{dx}\left[\frac{x^{3}}{3}-1\right]$$

Step 5: Differentiate the polynomial

Finally, we differentiate the polynomial: $$\frac{d}{dx}\left[\frac{x^{3}}{3}-1\right] = \frac{3x^{2}}{3} = x^{2}$$

Step 6: Combine all the derivatives

Now, we combine all the derivatives: $$\frac{dy}{dx} = \frac{1}{2\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}} \cdot \frac{1}{\sin\left(\frac{x^{3}}{3}-1\right)} \cdot \cos\left(\frac{x^{3}}{3}-1\right) \cdot x^{2}$$ $$\frac{dy}{dx} = \frac{x^{2} \cot\left(\frac{x^{3}}{3}-1\right)}{2\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}}$$

Final Answer: $\frac{x^{2} \cot\left(\frac{x^{3}}{3}-1\right)}{2\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}}$

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Pedagogical Audit

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

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a series of steps (chain rule, differentiation of logarithmic, trigonometric, and polynomial functions) 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 the chain rule, which are core concepts in the syllabus.