let ( g(x)=-2cos(x)+3x^{3} ). ( g(x)= )

Explanation:

Step1: Differentiate (-2\cos(x))

The derivative of (\cos(x)) is (-\sin(x)). Using the constant multiple rule, the derivative of (-2\cos(x)) is (-2\times(-\sin(x)) = 2\sin(x)).

Step2: Differentiate (3x^{3})

Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), for (n = 3) and the constant multiple rule, the derivative of (3x^{3}) is (3\times3x^{3-1}=9x^{2}).

Step3: Sum the derivatives

By the sum rule of differentiation ((u + v)^\prime=u^\prime+v^\prime), where (u=-2\cos(x)) and (v = 3x^{3}), we have (g^\prime(x)=2\sin(x)+9x^{2}).

Answer:

(2\sin(x)+9x^{2})