When writing rational expressions, you need to be aware that ...
1/4x = (1/4)x ≠ 1/(4x)
Parentheses around the denominator are required, unless you're typesetting the expression and can use a fraction bar for grouping.
The derivative of the curve expression is ...
y' = x - 1/(4x) . . . . . parentheses added to what you wrote
and the expression (1 -(y')^2) can be written ...
1 -(y')^2 = x^2 +1/2 +1/(16x^2) . . . . . parentheses added to what you wrote
The first and last terms of this trinomial are both perfect squares, so you might suspect the whole trinomial is a perfect square. You recall that ...
(a +b)^2 = a^2 + 2ab + b^2
This is a good "pattern" to remember. Using it is a matter of <em>pattern recognition</em>, as is the case with a lot of math.
Here, you have ...
a = x
b = 1/(4x)
In order for your trinomial to be a perfect square, the product 2ab must equal the middle term of your trinomial. (Spoiler: it does.)
2ab = 2(x)(1/(4x)) = (2x)/(4x) = 1/2 . . . . . matches the middle term of 1 -(y')^2
Hence your trinomial can be written as the square ...
1 -(y')^2 = (x +1/(4x))^2
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This is convenient because you want to integrate the square root of this. Your integral then becomes ...