Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), where \( h \neq 0 \), for the function below. \[ f(x)=\frac{1}{x-6} \] Simplify your answer as much as possible. \( \frac{f(x+h)-f(x)}{h}=\square \)

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = \frac{1}{x-6} \), we first need to evaluate \( f(x+h) \): \[ f(x+h) = \frac{1}{(x+h)-6} = \frac{1}{x+h-6} \] Now we can substitute \( f(x+h) \) and \( f(x) \) into the difference quotient: \[ \frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{x+h-6} - \frac{1}{x-6}}{h} \] To simplify the expression, we will combine the fractions in the numerator. We need a common denominator, which is \((x+h-6)(x-6)\): \[ \frac{1}{x+h-6} - \frac{1}{x-6} = \frac{(x-6) - (x+h-6)}{(x+h-6)(x-6)} = \frac{x-6 - x - h + 6}{(x+h-6)(x-6)} \] This simplifies to: \[ \frac{-h}{(x+h-6)(x-6)} \] Now plugging this back into the difference quotient gives: \[ \frac{-h}{(x+h-6)(x-6)} \cdot \frac{1}{h} = \frac{-1}{(x+h-6)(x-6)} \] Thus, the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = \frac{-1}{(x+h-6)(x-6)} \]