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Derivative of the inverse sine
The first derivative of the inverse sine is
Explication
We'll start with the sine, for which applies
x = sin y ⇔ y = arcsin x
Differentiation gives
Now we apply the chain rule
Turning over delivers
We want to trace this back to a function that only contains sin y, because we can then replace that with x, and in the conversion table we find
Substitution of y = arcsin x and sin y = x gives