Answer
The difference quotient for the given function is $-\dfrac{1}{x(x+h)}$
Work Step by Step
$f(x)=\dfrac{1}{x}$
Find the difference quotient $\dfrac{f(x+h)-f(x)}{h}$
Start by finding $f(x+h)$. Substitute $x$ by $x+h$ in $f(x)$ and simplify:
$f(x+h)=\dfrac{1}{x+h}$
Substitute the known values into the formula for the difference quotient and simplify:
$\dfrac{f(x+h)-f(x)}{h}=\dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}=\dfrac{\dfrac{x-(x+h)}{x(x+h)}}{h}=...$
$...=\dfrac{\dfrac{x-x-h}{x(x+h)}}{h}=\dfrac{\dfrac{-h}{x(x+h)}}{h}=-\dfrac{h}{xh(x+h)}=-\dfrac{1}{x(x+h)}$
The difference quotient for the given function is $-\dfrac{1}{x(x+h)}$
