find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=x/(x + 6) simplify…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{x + h}{(x + h)+6}=\frac{x + h}{x+h + 6}$

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

[ \begin{align*} f(x + h)-f(x)&=\frac{x + h}{x+h + 6}-\frac{x}{x + 6}\ &=\frac{(x + h)(x + 6)-x(x + h+6)}{(x + 6)(x+h + 6)}\ &=\frac{x^{2}+6x+hx + 6h-(x^{2}+hx+6x)}{(x + 6)(x+h + 6)}\ &=\frac{x^{2}+6x+hx + 6h - x^{2}-hx - 6x}{(x + 6)(x+h + 6)}\ &=\frac{6h}{(x + 6)(x+h + 6)} \end{align*} ]

Step3: Calculate the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{\frac{6h}{(x + 6)(x+h + 6)}}{h}\ &=\frac{6h}{(x + 6)(x+h + 6)}\times\frac{1}{h}\ &=\frac{6}{(x + 6)(x+h + 6)} \end{align*} ]

Answer:

$\frac{6}{(x + 6)(x+h + 6)}$