Find the derivative of sin⁻¹(x) with respect to x

To find the derivative of the inverse sine function, denoted as sin⁻¹(x), we can use a standard formula from calculus.

Derivative of sin⁻¹(x)

The derivative of sin⁻¹(x) with respect to x is given by:

Formula

f'(x) = 1 / √(1 - x²)

Domain

This derivative is valid for values of x in the interval:

• -1 ≤ x ≤ 1

Explanation

This means that as long as x is between -1 and 1, you can use this formula to find the slope of the tangent line to the curve of sin⁻¹(x) at any point.

Example

If you want to find the derivative at a specific point, say x = 0.5, you would substitute 0.5 into the formula:

f'(0.5) = 1 / √(1 - (0.5)²) = 1 / √(1 - 0.25) = 1 / √0.75 = 1 / (√3 / 2) = 2 / √3

This gives you the slope of the tangent line at that point on the graph of sin⁻¹(x).