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