Show me how to do the circled problem and include all steps

Derivatives. Fun, honestly. Anyways ...
Here's I would do it.

Take the radical out
1/(x+2)^(1/2)

Put it on top
(x+2)^(1/2)

Apply power rule
(1/2)(x+2)^((1/2)-1)
((x+2)^(-1/2))/2

Put it on the bottom
1/(2(x+2)^(1/2))

Put it back in radical form
1/(2√(x+2))

Hope I explained it well for you!