$\begin{array}{l}f(x)=\left(4 x^{2}+6\right)^{6}\left(3 x^{2}+6\right)^{11} \\ f^{\prime}(x)=\end{array}$

Step 1 ：Given the function \(f(x)=(4x^{2}+6)^{6}(3x^{2}+6)^{11}\), we are asked to find its derivative \(f'(x)\).

Step 2 ：We can use the product rule and the chain rule of differentiation to solve this. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. 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 ：Applying the product and chain rules, we find that \(f'(x)=48x(3x^{2}+6)^{11}(4x^{2}+6)^{5}+66x(3x^{2}+6)^{10}(4x^{2}+6)^{6}\).

Step 4 ：Thus, the derivative of the given function is \(\boxed{f'(x)=48x(3x^{2}+6)^{11}(4x^{2}+6)^{5}+66x(3x^{2}+6)^{10}(4x^{2}+6)^{6}}\).