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I can do the first step, which I think is to rationalize the numerator, but then I get stuck.

Then I get: \[\frac{ 1 }{ \sqrt{x + \Delta x } + \sqrt{x}} \]

lol i was writing instead of looking at what you wrote

now take the limit by replacing \(\Delta x\) by \(0\) and get
\[\frac{1}{2\sqrt{x}}\]

which happens to be the "DERIVATIVE" (yeah, i know) of \(\sqrt{x}\) a good one to memorize

Wow. Funny how the small steps confuse me, yet, I can do most of the "hard" things...

true, you did all the hard work
replacing \(\Delta x\) by zero is rather simple