What is the derivative of sin inverse(x)?

The derivative of sin inverse(x), often represented as arcsin(x), is derived from the relationship between the sine function and its inverse. To find the derivative, we can use implicit differentiation or recall the standard derivative formula for the arcsin function.

When we differentiate sin inverse(x), we find that the derivative is given by the formula:

[ \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}} ]

This formula is valid within the domain of the arcsin function, which is from -1 to 1. The expression (\sqrt{1 - x^2}) represents the length of the adjacent side in a right triangle where the hypotenuse is 1 and the opposite side is x. Since the arcsin function gives the angle whose sine is x, this geometrical relationship helps to establish the derivative.

Thus, when you see the derivative as ( \frac{1}{\sqrt{1 - x^2}} ), it correctly describes the rate at which the angle changes in relation to the value of x within the defined interval. This is why the choice indicating ( \frac{1}{\sqrt{1 - x^2