If $\lim\limits_{h\to 0} \dfrac{\arcsin(a+h)-\arcsin(a)}{h}=2$, which of the following could be the value of $a$ ?

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The definition of a derivative is give as the limit of the difference quotient:

$$\lim \limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

If we compare corresponding parts,

$$f(x) =\arcsin(a)$$

Using our knowledge of the derivative of the arcsin function:

$$f'(x)=\frac{1}{\sqrt{1-x^2}}$$

Setting this equal to $2$ and using the variable $a$:

$$2=\frac{1}{\sqrt{1-a^2}}$$ $$2\sqrt{1-a^2}=1$$ $$1-a^2=\frac{1}{4}$$ $$a^2=\frac{3}{4}$$ $$a= \pm \frac{\sqrt{3}}{2}$$