Find the derivative of the function $y=(4 x+5)^{4}(3 x+1)^{-2}$.

Step 1 ：Given the function \(y=(4 x+5)^{4}(3 x+1)^{-2}\)

Step 2 ：We can see that this function is a product of two functions, namely \((4x+5)^4\) and \((3x+1)^{-2}\)

Step 3 ：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 4 ：Let's find the derivatives of these two functions first

Step 5 ：The derivative of \((4x+5)^4\) can be found using the chain rule of differentiation, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, the outer function is \(u^4\) and the inner function is \(4x+5\). So, the derivative is \(16(4x + 5)^3\)

Step 6 ：Similarly, the derivative of \((3x+1)^{-2}\) can be found using the chain rule. Here, the outer function is \(u^{-2}\) and the inner function is \(3x+1\). So, the derivative is \(-6(3x + 1)^{-3}\)

Step 7 ：Applying the product rule, the derivative of the given function is \(16(4x + 5)^3(3x + 1)^{-2} - 6(4x + 5)^4(3x + 1)^{-3}\)

Step 8 ：Final Answer: The derivative of the function \(y=(4 x+5)^{4}(3 x+1)^{-2}\) is \(\boxed{16(4x + 5)^3(3x + 1)^{-2} - 6(4x + 5)^4(3x + 1)^{-3}}\)