The first derivative of the function $f(x)=\log _{4}\left(8^{x}+2^{x}\right)$ is given by
A. $3 \cdot \frac{8^{x}+2^{x}}{\ln (2)\left(8^{x}+2^{x}\right)}$
B. $\frac{8^{x} \ln (8)+2^{x} \ln (2)}{\left(8^{x}+2^{x}\right) \ln (4)}$
c. $\frac{1}{\left(8^{x}+2^{x}\right) \ln (4)}$
D. $4 \log _{3}\left(8 \cdot 7^{x}+2 \cdot 1^{x}\right)$
E. $\frac{3}{2} \cdot \frac{8^{x}+2^{x}}{\left(8^{x}+2^{x}\right)}$
F. $\frac{x 8^{x-1}+x 2^{x-1}}{\left(8^{x}+2^{x}\right) \ln (4)}$

Step 1 ：The function given is \(f(x)=\log _{4}\left(8^{x}+2^{x}\right)\).

Step 2 ：We need to find the derivative of this function.

Step 3 ：We can use the chain rule and the properties of logarithms to find the derivative.

Step 4 ：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 5 ：The properties of logarithms can be used to simplify the expression before taking the derivative.

Step 6 ：Applying the chain rule and the properties of logarithms, we get \(f'(x) = \frac{8^{x} \ln (8)+2^{x} \ln (2)}{\left(8^{x}+2^{x}\right) \ln (4)}\).

Step 7 ：This matches with option B in the question.

Step 8 ：Final Answer: The first derivative of the function \(f(x)=\log _{4}\left(8^{x}+2^{x}\right)\) is given by \(\boxed{\frac{8^{x} \ln (8)+2^{x} \ln (2)}{\left(8^{x}+2^{x}\right) \ln (4)}}\).