Answer
$$
y=f(x)=\frac{-6}{3x-5}\quad \text { from} \quad x=4\quad \text { to} \quad x=9
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=4 $ to $ b=9$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(9)-f(4)}{9-4}\\
& =\frac{(\frac{-6}{3(9)-5})-(\frac{-6}{3(4)-5})}{9-4}\\
& =\frac{(\frac{-6}{22})-(\frac{-6}{7})}{5}\\
& =\frac{(\frac{-21+66}{77})}{5}\\
& =\frac{(45)}{5(77)}\\
& =\frac{9}{77}\\
\end{split}
$$
and we can find that :
$$
y^{\prime}(x)=\frac{(3x-5)(0)-(-6)(3)}{(3x-5)^{2}}=\frac{18}{(3x-5)^{2}}
$$
Instantaneous rate of change at $x= 4$:
$$
y^{\prime}(4)=\frac{18}{(3(4)-5)^{2}}=\frac{18}{49}
$$
Work Step by Step
$$
y=f(x)=\frac{-6}{3x-5}\quad \text { from} \quad x=4\quad \text { to} \quad x=9
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=4 $ to $ b=9$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(9)-f(4)}{9-4}\\
& =\frac{(\frac{-6}{3(9)-5})-(\frac{-6}{3(4)-5})}{9-4}\\
& =\frac{(\frac{-6}{22})-(\frac{-6}{7})}{5}\\
& =\frac{(\frac{-21+66}{77})}{5}\\
& =\frac{(45)}{5(77)}\\
& =\frac{9}{77}\\
\end{split}
$$
and we can find that :
$$
y^{\prime}(x)=\frac{(3x-5)(0)-(-6)(3)}{(3x-5)^{2}}=\frac{18}{(3x-5)^{2}}
$$
Instantaneous rate of change at $x= 4$:
$$
y^{\prime}(4)=\frac{18}{(3(4)-5)^{2}}=\frac{18}{49}
$$
