Differentiability

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Using chain rule, d/dx[sin(x²)] equals

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cos(x²)

2x cos(x²)

x cos(x²)

2 cos(x²)

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About This Question

Subject: mathematics
Chapter: limit, continuity and differentiability
Topic: differentiability
Difficulty: Medium
Year: 2025

Solution

Correct Answer:

2x cos(x²)

Chain Rule Statement: $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) \cdot g^{'} (x)$ .
Identify Functions:
- Outer function $f (u) = sin u ⟹ f^{'} (u) = cos u$ .
- Inner function $u = g (x) = x^{2} ⟹ g^{'} (x) = 2 x$ .
Application:
- $\frac{d}{d x} [sin (x^{2})] = cos (x^{2}) \cdot \frac{d}{d x} (x^{2})$
- $= cos (x^{2}) \cdot (2 x)$
Result: $2 x cos (x^{2})$ .

This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of differentiability. It appeared in the 2025 exam.

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