Find the derivative. Find $f^{\prime}(x)$ for $f(x)=\left(4 x^{2}+3 x\right)^{2}$.

Step 1 ：Given the function \(f(x)=(4 x^{2}+3 x)^{2}\)

Step 2 ：This function is a composite function, where the outer function is \(u^2\) and the inner function is \(4x^2+3x\)

Step 3 ：The chain rule of differentiation states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function

Step 4 ：The derivative of the outer function \(u^2\) with respect to \(u\) is \(2u\)

Step 5 ：The derivative of the inner function \(4x^2+3x\) with respect to \(x\) is \(8x+3\)

Step 6 ：By applying the chain rule, the derivative of the given function is \(2(4x^2 + 3x)(8x + 3)\)

Step 7 ：Final Answer: The derivative of the function \(f(x)=(4 x^{2}+3 x)^{2}\) is \(\boxed{2(4x^2 + 3x)(8x + 3)}\)