f(x) = sin^-1 x
f'(x) = 1/√(1-x^2)
The function f(x) = sin^-1 x is the inverse function of the sine function with a restricted domain. Its domain is [-1, 1], and its range is [-π/2, π/2].
Graphically, the function f(x) = sin^-1 x is a curve that passes through the points (-1,-π/2), (0,0), and (1,π/2). It is a non-linear function that increases from -π/2 to π/2 as its input x increases from -1 to 1.
Some properties of the function f(x) = sin^-1 x that are worth mentioning are:
1. It returns an angle in radians between -π/2 and π/2 in response to an input value between -1 and 1.
2. The function is odd, which means f(-x)=-f(x). This property can be easily observed from the shape of the graph.
3. The derivative of f(x) = sin^-1 x can be determined using differentiation by parts, and it is given by:
f'(x) = 1 / sqrt(1 – x^2)
4. The integral of f(x) = sin^-1 x can also be determined using trigonometric substitution, and it is given by:
∫ sin^-1 x dx = x sin^-1 x + sqrt(1 – x^2) + C
where C is the constant of integration.
5. The function f(x) = sin^-1 x has a few real-world applications in physics, engineering, and mathematics. For example, it could be used to calculate the angle of incidence or reflection between two intersecting lines or surfaces.
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