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 \(a\):

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