What is the partial derivative with respect to $y$ of $f(x, y)=\log _{2}\left(4^{y}+2^{y}+2^{x}\right)$ ?
A. $\frac{2^{y}\left(2^{y} \ln (4)\right)+\ln (2)}{\ln (2)\left(4^{y}+2^{y}+2^{x}\right)}$
B. $\frac{2^{y}\left(2^{x+1}+1\right)}{4^{y}+2^{y}+2^{x}}$
c. $\frac{2^{y}\left(2^{y+1}\right)+1}{4^{y}+2^{y}+2^{x}}$
D. $\frac{4^{y} \ln (4)+2^{y} \ln (2)+2^{x} \ln (2)}{\left(4^{y}+2^{y}+2^{x}\right) \ln (2)}$
E. $\frac{4^{y} \ln (4)+2^{y} \ln (2)}{\left(4^{y}+2^{y}+2^{x}\right) \ln (2)}$

Step 1 ：The given function is \(f(x, y)=\log _{2}\left(4^{y}+2^{y}+2^{x}\right)\). We are asked to find the partial derivative of this function with respect to \(y\).

Step 2 ：To find the partial derivative of a function with respect to a variable, we differentiate the function with respect to that variable, treating all other variables as constants.

Step 3 ：In this case, we need to differentiate the function with respect to \(y\), treating \(x\) as a constant.

Step 4 ：The derivative of \(\log_b(a)\) with respect to \(a\) is \(\frac{1}{a \ln(b)}\).

Step 5 ：The derivative of \(a^b\) with respect to \(b\) is \(a^b \ln(a)\).

Step 6 ：Using these rules, we differentiate the function to get \(\frac{2^{y}\ln(2) + 4^{y}\ln(4)}{(2^{x} + 2^{y} + 4^{y})\ln(2)}\).

Step 7 ：Thus, the partial derivative of the function \(f(x, y)=\log _{2}\left(4^{y}+2^{y}+2^{x}\right)\) with respect to \(y\) is \(\boxed{\frac{4^{y} \ln (4)+2^{y} \ln (2)}{\left(4^{y}+2^{y}+2^{x}\right) \ln (2)}}\).