# What is the derivative of sin^{-1}(x)?

A derivative is the output resulting from differentiating a function with respect to its variable.

## Answer: The derivative of sin^{-1}x = dy/dx is 1/√(1−x^{2}), -1 < x < 1

Let us now see how to find the derivative using trigonometric identities.

**Explanation**:

Let us assume,

y = sin^{-1}(x) .........(i)

⇒ sin(y) = x

Differentiating with respect to y, we get

dx/dy = cos(y)

Thus,

dy/dx = 1/cos(y) .........(ii)

Substituting (i) in (ii),

We get, dy/dx = 1/cos(sin^{-1}(x))

Using the trignometric identity below and equation (i),

cos^{2}(y) = 1 - sin^{2}(y)

= 1 - (sin(sin^{-1}(x)))^{2}

= 1 - x^{2}

Thus, cos^{2}(y) = 1 - x^{2}

Taking square root on both the sides,

cos(y) = √(1−x^{2})

Plugging this into equation (ii), we finally get

dy/dx = 1/√(1−x^{2}), -1 < x < 1