Q601: If $xe^{y}=1$, then the value of $\frac{dy}{dx}$ at \(x=...

Class CBSE Class 12 Mathematics Derivatives Q #601

KNOWLEDGE BASED

UNDERSTAND

1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE

If $xe^{y}=1$, then the value of $\frac{dy}{dx}$ at $x=1$ is :

(A) -1

(B) 1

(D) $-\frac{1}{e}$

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AI Tutor Explanation

Detailed Solution

Step 1: Implicit Differentiation

Given the equation $xe^y = 1$, we need to find $\frac{dy}{dx}$. We will use implicit differentiation with respect to $x$.

Step 2: Differentiate both sides

Differentiating both sides of $xe^y = 1$ with respect to $x$, we get: $$\frac{d}{dx}(xe^y) = \frac{d}{dx}(1)$$ Using the product rule on the left side: $$\frac{d}{dx}(x) \cdot e^y + x \cdot \frac{d}{dx}(e^y) = 0$$ $$1 \cdot e^y + x \cdot e^y \cdot \frac{dy}{dx} = 0$$

Step 3: Solve for dy/dx

Now, we solve for $\frac{dy}{dx}$: $$e^y + xe^y \frac{dy}{dx} = 0$$ $$xe^y \frac{dy}{dx} = -e^y$$ $$\frac{dy}{dx} = -\frac{e^y}{xe^y}$$ $$\frac{dy}{dx} = -\frac{1}{x}$$

Step 4: Evaluate at x=1

We need to find the value of $\frac{dy}{dx}$ at $x=1$. $$\frac{dy}{dx}\Big|_{x=1} = -\frac{1}{1} = -1$$

Final Answer: -1

AI Suggestion: Option A

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of implicit differentiation and apply the product rule correctly to find the derivative.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply the procedure of implicit differentiation and the product rule to find the derivative.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of differentiation techniques, specifically implicit differentiation, which is a standard topic in the syllabus.