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I recently learned that relations like $f(x)=\sqrt{x}$ and $f(x)=\arcsin(x)$ are not actually functions but multivalued functions, since they take multiple outputs for a single input. So how come we can write formulas such as $$\frac{\mathrm{d}}{\mathrm{d}x} \big(\arcsin(x) \big)=\frac{1}{\sqrt{1-x^2}}$$, as far as I know, the derivative is defined as $f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$, which is defined for functions. We can also write $$\int_a ^b \sqrt{x}\mathrm{d}x$$, and it is well defined. My question is, what's going on? How can we apply a limit to something that isn't a function?

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The confusion happens when you do not distinguish between the functions $y= \sqrt x$ which is the positive branch of $x=y^2$ and the negative branch which is $y=-\sqrt x$

These are two different well defined functions.

For example $\sqrt {25}= 5$ and $-\sqrt {25}= -5$

For $arcsin(x)$ you pick the branch which satisfies $-\pi /2 \le \arcsin (x)\le \pi/2$ and that is well-defined.

For example $\arcsin ( 1/2)=\pi/6$

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