Differentiate. \[ y=\log _{4}\left(x^{4}+x\right) \]

Step 1 ：Given the function \(y=\log _{4}\left(x^{4}+x\right)\), we are asked to find its derivative.

Step 2 ：We can use the chain rule and the properties of logarithms to solve this problem. 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 ：The derivative of the logarithm base 4 is \(\frac{1}{x \ln(4)}\).

Step 4 ：The derivative of \(x^{4}+x\) is \(4x^{3}+1\).

Step 5 ：We multiply these two derivatives together to get the derivative of the original function.

Step 6 ：Thus, the derivative of the function \(y=\log _{4}\left(x^{4}+x\right)\) is \(\boxed{\frac{4x^{3}+1}{(x^{4}+x)\ln(4)}}\).