I don't know what you mean by first principles.
Think of f(x) as . Now use the power rule.
Oh ok. I see what you meant now. You are using the formula limit definition of a derivative.
You are on the right track and so close!
Like you said, you multiply everything by the conjugate of the numerator. Just simplify your last step.
Now let's simplify the numerator again. Find a common denominator for the two terms and combine. The common denominator is x(x+h). So...
.
Now when you simplify the numerator again, you'll notice you're left with a "-h" that can cancel with the "h" in the denominator leaving...
From here there is just a little more simplifying to go and you should get your final answer