One popular proof is to take $\sin{y} = x$ and then differentiate on both sides. But how do you prove it from first principles? Help very much appreciated.
First principle proof for derivatives of $\arcsin{x}$
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calculus
derivatives
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5Why isn't that proof from first principles? It's a general principle that if you know the derivative of a function, and it has an inverse, then you know the derivative of the inverse, just by the chain rule. – 2012-06-04