find the difference quotient of f; that is find (f(x + h)-f(x))/h, h≠0, for the function f(x)=6x/(x + 2)…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{6(x + h)}{(x + h)+2}=\frac{6x+6h}{x + h+2}$

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

[ \begin{align*} f(x + h)-f(x)&=\frac{6x + 6h}{x + h+2}-\frac{6x}{x + 2}\ &=\frac{(6x + 6h)(x + 2)-6x(x + h+2)}{(x + h+2)(x + 2)}\ &=\frac{6x^{2}+12x+6hx + 12h-(6x^{2}+6hx+12x)}{(x + h+2)(x + 2)}\ &=\frac{6x^{2}+12x+6hx + 12h - 6x^{2}-6hx-12x}{(x + h+2)(x + 2)}\ &=\frac{12h}{(x + h+2)(x + 2)} \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{\frac{12h}{(x + h+2)(x + 2)}}{h}\ &=\frac{12h}{h(x + h+2)(x + 2)}\ &=\frac{12}{(x + h+2)(x + 2)} \end{align*} ]

Answer:

$\frac{12}{(x + h+2)(x + 2)}$