Question

The differential coefficient of the  \( \sin(x^2 + 5) \) with respect to \( x \) will be:
 

• A. \( 2x \cos(x^2 + 5) \)
• B. \( 2x \sin(x^2 + 5) \)
• C. \( \cos(x^2 + 5) \)
• D. None of these

Hint:

For functions of the form \( \sin(f(x)) \), use the chain rule: \( \frac{d}{dx}[\sin(f(x))] = \cos(f(x)) \cdot f'(x) \).

Answer 1

The Correct Option is A

Let \( y = \sin(x^2 + 5) \). Differentiating with respect to \( x \): \[ \frac{dy}{dx} = \cos(x^2 + 5) \cdot \frac{d}{dx}(x^2 + 5). \] \[ \frac{dy}{dx} = \cos(x^2 + 5) \cdot 2x = 2x \cos(x^2 + 5). \] Hence, the correct answer is i) \( 2x \cos(x^2 + 5) \).