Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$, for the following function.
\[
f(x)=8 x+7
\]

Step 1 ：Given the function \(f(x)=8x+7\)

Step 2 ：We need to find the difference quotient of \(f\), that is, find \(\frac{f(x+h)-f(x)}{h}, h \neq 0\)

Step 3 ：Substitute \(f(x+h)\) and \(f(x)\) into the difference quotient formula: \(\frac{f(x+h)-f(x)}{h} = \frac{(8(x+h)+7)-(8x+7)}{h}\)

Step 4 ：Simplify the expression to get the difference quotient: \(\frac{8h}{h} = 8\)

Step 5 ：The difference quotient of the function \(f(x)=8x+7\) simplifies to 8. This means that the average rate of change of the function over the interval \(h\) is 8. This makes sense because the function is a linear function with a slope of 8, so the rate of change is constant and equal to the slope.

Step 6 ：Final Answer: The difference quotient of the function \(f(x)=8x+7\) is \(\boxed{8}\)