Derivative Notation: Let $f(x)=x$. Is $f′(x^2)=1$ or is $f′(x^2)=2x$? Let $f(x)=x$.

Is $f'(x^2)=1$ or is $f'(x^2)=2x$?

In other words, using this notation, are we evaluating the derivative at a $x^2$?
If you consider the function $f$ of a variable $x$, then $f'$ is likewise a function. When you write $f'(a)$, you evaluate that function at $a$. If $a$ happens to be $x^2$, then you evaluate at $x^2$. It has nothing to do with differentiating with respect to $a$ or $x^2$ or anything else. So, in your example, $f'(a)=1$ no matter what you put in for $a$ when $f(x)=x$.

If you consider $g(x)=x^2$ and the composition $(f\circ g)(x) = f(x^2)$, then
$(f\circ g)'(x) = f'(g(x))g'(x) = f'(x^2)\cdot 2x$.
As $f(x)=x$, then

$$\left(f(x^2)\right)'=(x^2)'.$$

The $'$ denotes differentiation on $x$.