The square root of 0 is undefined

Where is my mistake?

oh man, what are you talking about for a mess. baseless.

"I have to contradict you there. The e-function is an exception to the rule because e ^ x only takes positive values. I overlooked that in my reasoning earlier, I admit."

that's wrong again.

there are various possible definitions.
one of them has the power law for non-negative bases and real exponents. it depends on the base and exponent, not on the potency.

"f (x) = root (x ^ 2) either

a) f not defined for negative x or
b) f (x) = x (because: root (x ^ 2) = (x ^ 2) ^ (1/2) "=" x ^ (2 * 1/2) = x) "

no, you didn't understand something fundamental.
according to the power law: x² ^ (1/2) = x for x> 0
i.e. the function is undefined for x <= 0, for x> 0 it exists and is identical to x there
why they are not defined for x <0 or x <= 0 may be due to the bad properties of real numbers.
i didn't come up with this definition of real potency.

"What the problem is: the assumption that for x> = 0 root (x ^ 2) = x is obvious, but would first have to be proven in an exam situation - otherwise there is a risk of points being deducted (in your studies you would simply cross out the equals sign and the task would give a total of 0 points if the rest builds on it - from personal experience) "

basically yes, but the proof is trivial and is often not required.
(especially since the lecture or exercises deal with the basics and can be used)
there is also the possibility of anchoring this axiomatically, then you can give yourself proof.
it just depends on which premises are set up in the lecture.

"The first variant to derive the function according to the chain rule was the much safer and mathematically correct way. That the other one also works in this special case, I didn't think about that in the second"

this does not only apply in this special case, but also for every power as it is defined as a real-valued function

That this is the safer way is your subjective opinion, which is not and cannot be based on any mathematical facts.
I'm very pragmatic in this regard: he chose this route (probably the first), he solved it correctly (just not completely simplified), so it is the better route for him.
I would have solved it differently, in a way that was faster for my way of thinking.

"As far as the complex level is concerned, root (1) = 1 - the complex level only comes into consideration if you want to square root from negative numbers, e.g. root (-1) = i - imaginary unit, but does not belong here , is too high for the school material or not worth considering. "

that's wrong again.
apparently you have too little idea in the complex of topics.
the root is defined completely differently in the complex, it is not unambiguous there, incidentally, a certain mr gauss proved that around 1800, namely he showed: in C every non-constant polynomial breaks down into exactly n linear factors that do not necessarily have to be different. thus the n-th root is an n-element set - according to the def. the root in the complex.
it can even be shown that it is exactly n various n-th roots exist
you can write: 1 ^ (1/2) = e ^ (pi * i + pi * i * k) with k from Z.
you see: for every k in Z the following applies: Im (z) = 0
however, Re (z) alternates between 1 and -1.
the square root is therefore 2 definite, are two numbers.
for functional theory considerations one often says phi from [-pi / 2; pi / 2) to ensure uniqueness.
then a root would fall away.
the same can be done with -1, here is only the argument pi + 2pik (k from Z), therefore the sine term is not omitted, i.e. Im (z) <> 0.

At this point, the beautiful book function theory by fischer / lieb is recommended, from which I can quote (p.125):
We notice without proof that there can only be a continuous function f with (f (z)) ^ n = z (n> = 2) in a domain G subset C * if there is a logarithmic function on G [.. .].
Branches of the power function (with any exponent) exist in particular on simply connected areas that do not contain the zero point.
The power function that is defined on the plane cut along the negative real axis by means of the main branch of the logarithm is also called the main branch (or main value) of the power in question. For real b, the main branch of z ^ b on the positive real axis corresponds to the usual power x -> x ^ b

you see, they don't even speak of roots (not in the whole book), but of potencies.
In the complex, the root is just a small special case that does not need special consideration, actually there are not even roots there
In the real world it is a bit more complicated, precisely because the root is defined differently there and has to be defined differently.
this is a consequence of the algebraic incompleteness of R.

the mathematician's understanding of complex numbers goes far beyond your understanding, as you may see from the quote.
For mathematicians, the complex numbers are not necessarily there to be able to pull roots from negative numbers.
No, rather, with this auxiliary construct that can be obtained from R, mathematics can be viewed from a completely different point of view. so many things can be solved much more elegantly by introducing C.
and we you don't actually need roots from negative numbers, but use these numbers to define complex operators, and thus complex wave equations
those darn numbers appear everywhere ... (somehow I have to think of dirac matrices, for whatever reason)
if you say that it is not worth considering, you do not only shake my head, but also an estimated every function theorist.
this other way of ... thinking - I almost have to say - is very often worth considering, here too.
the only thing that speaks against this view is the school level here.
Hopefully I will still be forgiven for my finger gymnastics

"What do you mean by" you also find that you are using the law that you yourself stamp as invalid "? I used it for demonstration purposes in the example to show that it does not work, but otherwise not, right?"

Oh well...

which law do you use here:
"root (e ^ 2x / (4 * e ^ x))
= root (e ^ x / 4) "

In principle, only two are considered, and one can derive the other from one (with a few tools)

next time, please inform better before you fire sharp arrows wildly through the area.