“why does $x^{1/2} = +\sqrt{x}$ not $±\sqrt{x}$?”

Seems you are asking about, in the context of differentiation, you’re not asking about how to actually differentiate.

I think you’re asking why it is not true that x^{1/2} = ±\sqrt{x}?

Since it seems logical that it would be similar to this, \sqrt{x^{2}} = ±x

Basically, I would say it’s because you can write -x^{1/2}

If that’s what you want to say, you put the negative sign in front of it. If you want it to be positive, you don’t write the negative sign.

If you square ±\sqrt{x}

In either case, you will get x.

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