Answer
$\dfrac{f(x+h)-f(x)}{h}=-\dfrac{7}{(x+3)(x+h+3)}$
$\dfrac{f(x)-f(a)}{x-a}=-\dfrac{7}{(x+3)(a+3)}$
Work Step by Step
$f(x)=\dfrac{7}{x+3}$
$\textbf{Evaluate}$ $\dfrac{f(x+h)-f(x)}{h}$
First, substitute $x$ by $x+h$ in the given function and simplify to find $f(x+h)$:
$f(x+h)=\dfrac{7}{x+h+3}$
Substitute $f(x+h)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x+h)-f(x)}{h}=\dfrac{\dfrac{7}{x+h+3}-\dfrac{7}{x+3}}{h}=...$
$...=\dfrac{\dfrac{7(x+3)-7(x+h+3)}{(x+3)(x+h+3)}}{h}=...$
$...=\dfrac{7x+21-7x-7h-21}{h(x+3)(x+h+3)}=-\dfrac{7h}{h(x+3)(x+h+3)}=...$
$...=-\dfrac{7}{(x+3)(x+h+3)}$
$\textbf{Evaluate}$ $\dfrac{f(x)-f(a)}{x-a}$
Substitute $x$ by $a$ in $f(x)$ to find $f(a)$:
$f(a)=\dfrac{7}{a+3}$
Substitute $f(a)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x)-f(a)}{x-a}=\dfrac{\dfrac{7}{x+3}-\dfrac{7}{a+3}}{x-a}=\dfrac{\dfrac{7(a+3)-7(x+3)}{(x+3)(a+3)}}{x-a}=...$
$...=\dfrac{7a+21-7x-21}{(x-a)(x+3)(a+3)}=\dfrac{7a-7x}{(x-a)(x+3)(a+3)}=...$
Take out common factor $-7$ from the numerator and simplify again:
$...=-\dfrac{7(x-a)}{(x-a)(x+3)(a+3)}=-\dfrac{7}{(x+3)(a+3)}$
