Using the definition of the derivative, we have
[tex]\displaystyle f'(11) = \lim_{x\to11} \frac{\sqrt{x-2} - \sqrt{11-2}}{x - 11} = \lim_{x\to11} \frac{\sqrt{x-2} - 3}{x - 11}[/tex]
Rationalize the numerator.
[tex]\displaystyle \frac{\sqrt{x-2}-3}{x-11} \cdot \frac{\sqrt{x-2}+3}{\sqrt{x-2}+3} = \frac{\left(\sqrt{x-2}\right)^2 - 3^2}{(x-11)\left(\sqrt{x-2}+3\right)} = \frac{x - 11}{(x - 11) \left(\sqrt{x-2} + 3\right)}[/tex]
If [tex]x\neq11[/tex], we can simplify the limit to
[tex]\displaystyle f'(11) = \lim_{x\to11} \frac{x-11}{(x-11)\left(\sqrt{x-2}+3\right)} = \lim_{x\to11} \frac1{\sqrt{x-2} + 3}[/tex]
The function in the remaining limit is continuous at [tex]x=11[/tex], so we end up with
[tex]\displaystyle f'(11) = \frac1{\sqrt{11-2} + 3} = \boxed{\frac16}[/tex]
