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