This is not particularly easy using the limit definition, so that's fine. Using the limit definition:
#lim_(h->0) [f(x+h) - f(x)]/h#
Plug it in:
#= lim_(h->0) [1/(sqrt(x+h-2)) - 1/(sqrt(x-2))]/h#
Cross-multiply:
#= lim_(h->0) [sqrt(x-2)/(sqrt(x+h-2)sqrt(x-2)) - sqrt(x+h-2)/(sqrt(x-2)sqrt(x+h-2))]/h#
Combine:
#= lim_(h->0) ([sqrt(x-2) - sqrt(x+h-2)]/(sqrt(x+h-2)sqrt(x-2)))/h#
Move stuff around:
#= lim_(h->0) [sqrt(x-2) - sqrt(x+h-2)]/(h(sqrt(x+h-2)sqrt(x-2)))#
Multiply by the complex conjugate:
#= lim_(h->0) [sqrt(x-2) - sqrt(x+h-2)]/(h(sqrt(x+h-2)sqrt(x-2)))*(sqrt(x-2) + sqrt(x+h-2))/(sqrt(x-2) + sqrt(x+h-2))#
#= lim_(h->0) [x-2 - (x+h-2)]/(h(sqrt(x+h-2)sqrt(x-2))*(sqrt(x-2) + sqrt(x+h-2)))#
#= lim_(h->0) [-cancel(h)]/(cancel(h)(sqrt(x+h-2)sqrt(x-2))*(sqrt(x-2) + sqrt(x+h-2)))#
#= -lim_(h->0) 1/[(sqrt(x+h-2)sqrt(x-2))*(sqrt(x-2) + sqrt(x+h-2)))#
Hey, that looks nice now. You didn't even have to multiply out the denominator yet. So plugging in #h = 0#:
#= -lim_(h->0) 1/[(sqrt(x-2)sqrt(x-2))*(sqrt(x-2) + sqrt(x-2)))#
#= -lim_(h->0) 1/[(x-2)*(2sqrt(x-2)))#
#= color(blue)(-1/[2(x-2)^("3/2")])#
If you took the actual power rule derivative, you would get:
#d/(dx)[(x-2)^(-"1/2")] = -1/2*(x-2)^(-"3/2") = color(blue)(-1/[2(x-2)^("3/2")])#
