Question

The differentiation of the \(\sin(x^2 + 1)\) function with respect to \(x\), will be

• A. \(\sin x \cdot \cos(x^2 + 1)\)
• B. \(x^2 \cos(x^2 + 1)\)
• C. \(x \cdot \cos(x^2 + 1)\)
• D. \(2x \cdot \cos(x^2 + 1)\)

Hint:

Chain rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).
Always differentiate the outer function first, then multiply by the derivative of the inner function.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the chain rule of differentiation.

Step 2: Key Formula or Approach:
Chain rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Step 3: Detailed Explanation:
Given function: \(y = \sin(x^2 + 1)\)
Let \(u = x^2 + 1\)
\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
\(\frac{dy}{du} = \cos u = \cos(x^2 + 1)\)
\(\frac{du}{dx} = 2x\)
\(\frac{dy}{dx} = \cos(x^2 + 1) \cdot 2x = 2x \cos(x^2 + 1)\)
Thus, the derivative is \(2x \cos(x^2 + 1)\).

Step 4: Final Answer:
Thus, the differentiation of \(\sin(x^2 + 1)\) is \(2x \cos(x^2 + 1)\).
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