Find the difference quotient $\frac{f(x+h)-f(x)}{h}$, where $h \neq 0$, for the function below. \[ f(x)=\frac{1}{x-8} \] Simplify your answer as much as possible. \[ \frac{f(x+h)-f(x)}{h}=\square \]

Step 1 ：The difference quotient is a measure of the average rate of change of the function over the interval h. To find the difference quotient for the function \(f(x)=\frac{1}{x-8}\), we need to substitute \(x+h\) into the function for \(x\), subtract the original function, and then divide by \(h\).

Step 2 ：Substitute \(x+h\) into the function for \(x\): \(f(x+h) = \frac{1}{(x+h)-8}\)

Step 3 ：Subtract the original function: \(f(x+h) - f(x) = \frac{1}{(x+h)-8} - \frac{1}{x-8}\)

Step 4 ：Divide by \(h\): \(\frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{(x+h)-8} - \frac{1}{x-8}}{h}\)

Step 5 ：Simplify the expression to get the difference quotient: \(\frac{f(x+h)-f(x)}{h} = -\frac{1}{(x - 8)(h + x - 8)}\)

Step 6 ：Final Answer: \(\boxed{-\frac{1}{(x - 8)(h + x - 8)}}\)