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