Class CBSE Class 12 Mathematics Derivatives Q #1698
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $e^{-x}+e^{-y}=2$, then $\frac{dy}{dx}$ is
(A) $e^{x-y}$
(B) $e^{y-x}$
(C) $-e^{x-y}$
(D) $-e^{y-x}$
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Correct Answer: D

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Detailed Solution

Step 1: Differentiate the given equation

Given the equation $e^{-x} + e^{-y} = 2$. Differentiating both sides with respect to $x$, we apply the chain rule:

$$\frac{d}{dx}(e^{-x}) + \frac{d}{dx}(e^{-y}) = \frac{d}{dx}(2)$$ $$-e^{-x} - e^{-y} \cdot \frac{dy}{dx} = 0$$

Step 2: Isolate the derivative term

Rearrange the equation to solve for $\frac{dy}{dx}$:

$$-e^{-y} \cdot \frac{dy}{dx} = e^{-x}$$ $$\frac{dy}{dx} = -\frac{e^{-x}}{e^{-y}}$$

Step 3: Simplify the expression

Using the laws of exponents, specifically $\frac{e^a}{e^b} = e^{a-b}$:

$$\frac{dy}{dx} = -e^{-x - (-y)}$$ $$\frac{dy}{dx} = -e^{y-x}$$

Final Answer: $-e^{y-x}$

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the chain rule of differentiation and algebraic manipulation of exponential functions to solve for the derivative.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific sequence of mathematical steps (differentiation, isolation, and simplification) to reach the result.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to handle implicit differentiation, a core competency in the Calculus unit of the Mathematics curriculum.

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