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