Question about differentials?

I have noticed that when mathematicians state a second derivative, they use the notation d^2y/dx^2. Is the square root of dx^2 equal to dx? Likewise, what is the square root of d^2y. My dad says the reason the ^2 comes after the d instead of the y is because y is a function and not a free variable, so wouldn't they be the same. Also, does the squared after the differentials mean that it's actually squared, or is that just the notation used to denote a second differential?