For a simple equation $y=x^2$, when we take derivative on both side with respect to $x$ we get
$dy/dx = 2x$
$dy = 2xdx$
All of a sudden, the second equation is like a trick, we can skip the intermediate step and can put $d$ directly on both sides of the equation $y=x^2$ and we can still get $dy = 2xdx$, it follows the rule "Take differentiation on both sides of the function with respect to its own variable".
Can I interpret $dy = 2xdx$ as:
the change on the left side equals to the change on the right side?
If this interpretation is valid and not just a math trick I would prefer to use this "rule" rather than take derivative on both sides and multiply $dx$ to have the same result.
So everytime when I want the tiny change of y equals to the tiny change of $x^2$, I can directly write down
$dy = 2xdx$
Think of a square whose side is $x$, and increment each side by a magnitude $dx$. Then, $dy$ will be the change in the area of the square. If you make a drawing, you will see that to the first order in $dx$, $dy =2xdx$.