Answer
The difference quotient for the provided function is \[-\frac{1}{2x\left( x+h \right)}\].
Work Step by Step
Consider the provided function: $f\left( x \right)=\frac{1}{2x}$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$f\left( x+h \right)=\frac{1}{2\left( x+h \right)}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{\frac{1}{2\left( x+h \right)}-\frac{1}{2x}}{h} \\
& =\frac{\frac{x-\left( x+h \right)}{2x\left( x+h \right)}}{h} \\
& =\frac{-h}{2hx\left( x+h \right)} \\
& =-\frac{1}{2x\left( x+h \right)}
\end{align}$
Hence, the difference quotient for the provided function is $-\frac{1}{2x\left( x+h \right)}$.
