Answer
The difference quotient for the given function is $\dfrac{1}{\sqrt{x+h-1}+\sqrt{x-1}}$
Work Step by Step
$f(x)=\sqrt{x-1}$
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:
$f(x+h)=\sqrt{x+h-1}$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{\sqrt{x+h-1}-\sqrt{x-1}}{h}=...$
Rationalize the numerator and simplify:
$...=\dfrac{\sqrt{x+h-1}-\sqrt{x-1}}{h}\cdot\dfrac{\sqrt{x+h-1}+\sqrt{x-1}}{\sqrt{x+h-1}+\sqrt{x-1}}=...$
$...=\dfrac{(\sqrt{x+h-1})^{2}-(\sqrt{x-1})^{2}}{h(\sqrt{x+h-1}+\sqrt{x-1})}=...$
$...=\dfrac{x+h-1-x+1}{h(\sqrt{x+h-1}+\sqrt{x-1})}=\dfrac{h}{h(\sqrt{x+h-1}+\sqrt{x-1})}=...$
$...=\dfrac{1}{\sqrt{x+h-1}+\sqrt{x-1}}$
The difference quotient for the given function is $\dfrac{1}{\sqrt{x+h-1}+\sqrt{x-1}}$
