Calculate the derivative of the function. \[ \begin{array}{r} g(x)=\left(2 x^{2}+x+9\right)^{-3} \\ g^{\prime}(x)=\square \end{array} \] Submit Answer

Step 1 ：Given the function \(g(x) = (2x^2 + x + 9)^{-3}\), we are asked to find its derivative.

Step 2 ：To find the derivative of the function, we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 3 ：In this case, the outer function is \(f(u) = u^{-3}\) and the inner function is \(u = 2x^2 + x + 9\).

Step 4 ：The derivative of the function \(g(x) = (2x^2 + x + 9)^{-3}\) is \(g^{\prime}(x) = \frac{-12x - 3}{(2x^2 + x + 9)^4}\).

Step 5 ：\(\boxed{g^{\prime}(x) = \frac{-12x - 3}{(2x^2 + x + 9)^4}}\) is the final answer.