A Comprehensive Guide on Finding the Derivative of the Inverse Sine Function (arcsin(x)) and its Range

d/dx arcsin(x)

To find the derivative of a function, let’s start with the inverse sine function, arcsin(x)

To find the derivative of a function, let’s start with the inverse sine function, arcsin(x).

The inverse sine function, also denoted as arcsin(x) or sin^(-1)(x), is the inverse of the sine function. It is defined for values of x between -1 and 1, inclusive, and returns angles in the range of -π/2 to π/2 radians (-90° to 90°). The output of arcsin(x) is the angle whose sine is x.

Now, to find the derivative of arcsin(x), we use the chain rule of differentiation. Recall that the chain rule states that if we have a composite function y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x).

In our case, y = arcsin(x), so we can view it as y = f(g(x)), where f(u) = arcsin(u) and g(x) = x.

Let’s differentiate f(u) = arcsin(u) with respect to u first. The derivative of arcsin(u) with respect to u is given by (1/sqrt(1 – u^2)). We can verify this using trigonometric identities or by using implicit differentiation.

Now, using the chain rule, we can find the derivative of y = arcsin(x) with respect to x:

dy/dx = f'(g(x)) * g'(x)
= (1/sqrt(1 – (g(x))^2)) * (d/dx(g(x)))

Substituting g(x) = x, we have:

dy/dx = (1/sqrt(1 – x^2)) * (d/dx(x))
= (1/sqrt(1 – x^2)) * 1
= 1/sqrt(1 – x^2)

Therefore, the derivative of arcsin(x) with respect to x is 1/sqrt(1 – x^2).

This result indicates that the derivative of arcsin(x) is the reciprocal of the square root of (1 – x^2). This value is only defined for x between -1 and 1, as arcsin(x) is not defined outside that range.

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