Derivative of $y =sin^{-1}(frac{a}{x})$ w.r.t. x

Question

I am unsure of this derivative. The answer should apparently be $frac{-a}{xsqrt{x^2-a^2}}$ by using the chain rule. However I arrived at a slightly different answer using implicit diff of $frac{-a}{x^2sqrt{1-frac{a^2}{x^2}}}$. I arrived at this answer by noticing $cos(sin^{-1}(frac{a}{x})) = sqrt{1-frac{a^2}{x^2}}$ by forming a right angled triangle with hypotenuses of 1.
The two derivatives agree on positive values for x, but they have opposite signs for negative values of x (surely mine is correct since at negative values of x the gradient should be negative not positive)?

Äres · Answer

Your answer is correct. The given answer is only valid for $x>0$ in the denominator and it should have been $|x|=sqrt{x^{2}}$ instead of x as mentioned by @Ninad Munshi.
Alternatively applying the differentiation rule:
$$frac{d}{dx}arcsin(u(x))=frac{1}{sqrt{1-u^{2}(x)}}u'(x)$$ and the chain rule we obtain that $$frac{d}{dx}arcsin(frac{a}{x})=frac{frac{d}{dx}(frac{a}{x})}{sqrt{1-(frac{a}{x})^{2}}}=-frac{a}{x^{2}sqrt{1-frac{a^{2}}{x^{2}}}}.$$

Derivative of $y =sin^{-1}(frac{a}{x})$ w.r.t. x

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