Answer 1

You can try using taylor series to calculate more precise square roots.

https://www.sololearn.com/discuss/292240/?ref=app

The math remains the same.

Answer 2

You can read the formula from here : https://www.mathsisfun.com/algebra/taylor-series.html

The derivative of √x is 1/(2√x)

Answer 3

Here is a better process, without derivatives : burningmath.blogspot.com/2013/12/finding-square-roots-of-numbers-that.html?m=1

Answer 4

you can always use the newton method:

https://en.m.wikipedia.org/wiki/Newton%27s_method