find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=\fra…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{x + h}{(x + h)+9}=\frac{x + h}{x+h + 9}$

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=\frac{x + h}{x+h + 9}-\frac{x}{x + 9}\ &=\frac{(x + h)(x + 9)-x(x + h + 9)}{(x + 9)(x+h + 9)}\ &=\frac{x^{2}+9x+hx + 9h-(x^{2}+hx+9x)}{(x + 9)(x+h + 9)}\ &=\frac{x^{2}+9x+hx + 9h - x^{2}-hx - 9x}{(x + 9)(x+h + 9)}\ &=\frac{9h}{(x + 9)(x+h + 9)} \end{align*} ]

Step3: Calculate the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{\frac{9h}{(x + 9)(x+h + 9)}}{h}\ &=\frac{9h}{(x + 9)(x+h + 9)}\times\frac{1}{h}\ &=\frac{9}{(x + 9)(x+h + 9)} \end{align*} ]

Answer:

$\frac{9}{(x + 9)(x+h + 9)}$