Class CBSE Class 12 Mathematics Derivatives Q #1697
KNOWLEDGE BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $\sin^{-1}x=y$ then $\frac{dy}{dx}$ is:
(A) $\cos^{-1}x$
(B) $\cos y$
(C) $\frac{1}{1-x^{2}}$
(D) $\sec y$
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Correct Answer: D

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Detailed Solution

Step 1: Express the equation in terms of x

Given the equation sin-1x = y, we can rewrite this by taking the sine of both sides to isolate x: $$x = \sin y$$

Step 2: Differentiate with respect to y

Differentiate both sides of the equation x = sin y with respect to y: $$\frac{dx}{dy} = \frac{d}{dy}(\sin y) = \cos y$$

Step 3: Apply the reciprocal rule

To find dy/dx, we use the property of derivatives: $$\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$$ Substituting the result from Step 2: $$\frac{dy}{dx} = \frac{1}{\cos y}$$

Step 4: Simplify the expression

Using the trigonometric identity sec y = 1/cos y, we get: $$\frac{dy}{dx} = \sec y$$

Final Answer: sec y

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the chain rule or implicit differentiation techniques to inverse trigonometric functions.
Knowledge Dimension: PROCEDURAL
Justification: The question tests the student's ability to execute a specific mathematical algorithm (differentiation of inverse functions) rather than just recalling a formula.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the core curriculum requirement regarding the derivatives of inverse trigonometric functions found in Chapter 5 (Continuity and Differentiability).

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