How To Find The Derivative Of Arcsin(X) Using Inverse Function Rule: Step-By-Step Guide

derivative of sin⁻¹x

derivative of sin⁻¹x is equal to 1 divided by the square root of (1-x²).

The derivative of sin⁻¹x can be determined by using the formula for the derivative of inverse functions, which is:

d/dx [f⁻¹(x)] = 1 / f ‘(f ⁻¹(x))

Applying this formula to sin⁻¹x, we get:

d/dx [sin⁻¹x] = 1 / cos (sin⁻¹x)

To simplify this expression, we can use the Pythagorean identity for the sine and cosine functions, which is:

sin²θ + cos²θ = 1

Rearranging this equation, we get:

cos²θ = 1 – sin²θ

Substituting θ = sin⁻¹x into this equation, we get:

cos²(sin⁻¹x) = 1 – sin²(sin⁻¹x)

Using the definition of the inverse sine function, we can rewrite sin(sin⁻¹x) as x, and cos(sin⁻¹x) as √(1-x²). Therefore,

cos²(sin⁻¹x) = 1 – x²

Substituting this expression into the previous equation for the derivative of sin⁻¹x, we get:

d/dx [sin⁻¹x] = 1 / √(1-x²)

Therefore, the derivative of sin⁻¹x is equal to 1 divided by the square root of (1-x²).

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