Q1268: If $f(x)=|tan~2x|$, then find the value of $f^{\prime}(x)$ a...

Class CBSE Class 12 Mathematics Derivatives Q #1268

COMPETENCY BASED

UNDERSTAND

2 Marks 2024 AISSCE(Board Exam) VSA

If $f(x)=|tan~2x|$, then find the value of $f^{\prime}(x)$ at $x=\frac{\pi}{3}$

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AI Tutor Explanation

Detailed Solution

Step 1: Analyze the function

Given $f(x) = |\tan 2x|$. We need to find $f'(x)$ at $x = \frac{\pi}{3}$. First, we need to determine if $\tan 2x$ is positive or negative at $x = \frac{\pi}{3}$.

Step 2: Determine the sign of tan(2x) at x = pi/3

At $x = \frac{\pi}{3}$, $2x = \frac{2\pi}{3}$. Since $\frac{2\pi}{3}$ is in the second quadrant, $\tan \frac{2\pi}{3}$ is negative. Therefore, $\tan 2x < 0$ at $x = \frac{\pi}{3}$.

Step 3: Rewrite the function without absolute value

Since $\tan 2x < 0$ at $x = \frac{\pi}{3}$, we can rewrite $f(x)$ as $f(x) = -\tan 2x$ in a neighborhood around $x = \frac{\pi}{3}$.

Step 4: Differentiate the function

Now, we differentiate $f(x) = -\tan 2x$ with respect to $x$: $$f'(x) = -\frac{d}{dx}(\tan 2x) = -(\sec^2 2x) \cdot 2 = -2\sec^2 2x$$

Step 5: Evaluate the derivative at x = pi/3

Now, we evaluate $f'(x)$ at $x = \frac{\pi}{3}$: $$f'\left(\frac{\pi}{3}\right) = -2\sec^2\left(2 \cdot \frac{\pi}{3}\right) = -2\sec^2\left(\frac{2\pi}{3}\right)$$ Since $\sec\left(\frac{2\pi}{3}\right) = \frac{1}{\cos\left(\frac{2\pi}{3}\right)} = \frac{1}{-\frac{1}{2}} = -2$, we have: $$f'\left(\frac{\pi}{3}\right) = -2(-2)^2 = -2(4) = -8$$

Final Answer: -8

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the properties of absolute value functions and trigonometric functions to correctly differentiate the given function. They must also understand in which quadrant the angle lies to determine the sign of the trigonometric function.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply the rules of differentiation to a trigonometric function involving an absolute value. This involves a series of steps, including determining the sign of the function inside the absolute value, rewriting the function, and then applying the chain rule.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of differentiation and trigonometric functions in a problem-solving context, requiring them to analyze the function and choose the correct approach.