The derivative of f(x) = sin(x^3 + 6) with respect to x is

The correct answer is Option (3) → \(3x^{2}\cos(x^{3} + 6)\)

Let \(u = x^{3} + 6\), so \(f = \sin u\). By the chain rule, the derivative of the outer function is multiplied by the derivative of the inner one:

    \( \displaystyle f'(x) = \cos u \cdot \frac{du}{dx} \)

    \( = \cos(x^{3} + 6) \cdot 3x^{2} \)

    \( = 3x^{2}\cos(x^{3} + 6) \)