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

We will solve it using trigonometric identity sin^{2}y + cos^{2}y = 1.

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

Let see, how we can solve using this trigonometric identity.

**Explanation:**

We have to find the derivative of cos^{-1}x.

Let us assume y = cos^{-1}x. Then, cos y = x

Differentiate cos y = x implicitly with respect to x.

(-sin y) dy/dx = 1 ----- (i)

By trigonometric identity, we know that

sin^{2}y + cos^{2}y = 1

⇒ sin^{2}y + x^{2 }= 1

⇒ sin^{2}y = 1 – x^{2}

⇒ sin y = √(1 − x^{2})

Substituting the above value in (i), we get

−√(1 − x^{2})^{ }dy/dx = 1

⇒ dy/dx = –1/√(1 − x^{2}), -1 < x < 1