# What is the derivative of #f(x)=cos^-1(x)# ?

##### 2 Answers

#### Explanation:

In general,

Here's how we obtain this common derivative:

Differentiate both sides of

This will entail using Implicit Differentiation on the right side:

Solve for

We need to get rid of the

We previously said

Now, recall the identity

In the identity, replace

Thus,

#f(x)=cos^-1(x)" "=>" "cos(f(x))=x#

Take the derivative of both sides. Use the chain rule on the left.

#-sin(f(x))*f'(x)=1#

#=>" "f'(x)=(-1)/sin(f(x))=(-1)/sqrt(1-cos^2(f(x)))#

The last step came from the identity

#f'(x)=(-1)/sqrt(1-x^2)#

Note about domain: the domain of