Find the derivative of the function $y=(3 x+4)^{4}(4 x+1)^{-3}$. \[ \frac{d y}{d x}= \]

Step 1 ：Given the function \(y=(3x+4)^{4}(4x+1)^{-3}\).

Step 2 ：We need to find the derivative of this function. We can use the product rule of differentiation which 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.

Step 3 ：Let's denote the first function as \(f=(3x+4)^4\) and the second function as \(g=(4x+1)^{-3}\).

Step 4 ：First, we find the derivative of \(f\), denoted as \(f'\). Using the chain rule, we get \(f'=12(3x+4)^3\).

Step 5 ：Next, we find the derivative of \(g\), denoted as \(g'\). Using the chain rule, we get \(g'=-12(4x+1)^{-4}\).

Step 6 ：Now, we can apply the product rule. The derivative of the function \(y\) is \(y'=f'g+fg'\). Substituting the expressions for \(f\), \(f'\), \(g\), and \(g'\), we get \(y'=-12(3x+4)^4(4x+1)^{-4} + 12(3x+4)^3(4x+1)^{-3}\).

Step 7 ：\(\boxed{y'=-12\frac{(3x+4)^4}{(4x+1)^4} + 12\frac{(3x+4)^3}{(4x+1)^3}}\) is the derivative of the given function.