consider the function $f(x)=\frac{-1}{-6x + 9}$. simplify the difference quotient. $\frac{f(x + h)-f(x)}{h}$

Question

Accepted Answer

# Explanation:
## Step1: Find $ f(x + h) $
Given $ f(x)=\frac{-1}{-6x + 9}=\frac{1}{6x-9} $, then $ f(x + h)=\frac{1}{6(x + h)-9}=\frac{1}{6x+6h - 9} $

## Step2: Substitute $ f(x + h) $ and $ f(x) $ into the difference - quotient formula
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{\frac{1}{6x + 6h-9}-\frac{1}{6x-9}}{h}\
&=\frac{\frac{(6x - 9)-(6x + 6h-9)}{(6x + 6h-9)(6x-9)}}{h}\
\end{align*}
$$

## Step3: Simplify the numerator of the big - fraction
$$
\begin{align*}
(6x - 9)-(6x + 6h-9)&=6x-9 - 6x-6h + 9\
&=-6h
\end{align*}
$$

## Step4: Simplify the entire expression
$$
\begin{align*}
\frac{\frac{-6h}{(6x + 6h-9)(6x-9)}}{h}&=\frac{-6h}{(6x + 6h-9)(6x-9)}\cdot\frac{1}{h}\
&=\frac{-6}{(6x + 6h-9)(6x-9)}
\end{align*}
$$

# Answer:
$\frac{-6}{(6x + 6h-9)(6x-9)}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \( f(x + h) \)

Step2: Substitute \( f(x + h) \) and \( f(x) \) into the difference - quotient formula

Step3: Simplify the numerator of the big - fraction

Step4: Simplify the entire expression

Answer: