Difference between differentiation notations

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Please is there a difference between:

$\frac{dy}{dx}$ of $(1 + x^2)$ where $y = (1 + x^2)$

and

$\frac{d(1+x^2)}{dx}$

Aren't they both going to result in $2x$?

Just curious as to why in some calculations its written one way and in some, the other way.