Find the derivative of the function. \[ r(t)=\frac{1}{2}\left(8 t^{2}+t\right)^{-3} \] \[ f^{\prime}(t)= \]

Step 1 ：The problem is asking for the derivative of the function \(r(t)=\frac{1}{2}(8t^2+t)^{-3}\).

Step 2 ：To find the derivative of this 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 \(f(x) = \frac{1}{2}x^{-3}\) and the inner function is \(g(x) = 8t^2 + t\). So, we need to find the derivatives of these two functions and then multiply them together.

Step 4 ：The derivative of the function \(r(t)=\frac{1}{2}(8t^2+t)^{-3}\) is \(f^{\prime}(t)=\frac{-48t - 3}{2(8t^2 + t)^4}\).

Step 5 ：\(\boxed{f^{\prime}(t)=\frac{-48t - 3}{2(8t^2 + t)^4}}\) is the final answer.