Derivative question -  x^4 + 1/16x^4 to x^4 - 1/2 +1/16x^4?

What's going on is that you added 1 to (x^2 - (1/(4x^2))^2.  This is why it's important that you added those steps

(x^2 - 1/(4x^2))^2 =>

x^4 - 2 * x^2 * 1/(4x^2) + 1/(16x^4) =>

x^4 - (2/4) * x^2/x^2 + 1/(16x^4) =>

x^4 - (1/2) * 1 + 1/(16x^4) =>

x^4 - (1/2) + 1/(16x^4)

At no point did anything say that the first derivative of x^4 - 1/(4x^2) is x^4 - 1/2 + 1/(16x^4).  Words are important.  Use them correctly.

So, when you add 1 to x^4 - 1/2 + 1/(16x^4), you get

x^4 + 1/2 + 1/(16x^4) =>

(x^2)^2 + 2 * x^2 * 1/(4x^2) + (1/(4x^2))^2 =>

(x^2 + 1/(4x^2))^2

Nothing happening here is anything more complicated than

(x + y)^2 = x^2 + 2xy + y^2

or

(x - y)^2 = x^2 - 2xy + y^2

I guess this may have been an arc-length question?

That would be the reason for calculating sqrt[1 + (y')^2],

so if y' were x^4 - 1/(4x)^2,

the sqrt[1 + (y')^2] would indeed come out to x^4 + 1/(4x)^2.