If f is a bijection satisfying `f'(x)=sqrt((1-{f(x)}^(2))`, then `(f^(1))'x`

Options

A. is equal to `(1)/(sqrt(1-x^(2))`
B. may not exist for every `x in R`
C. may not be known explicitly
D. is equal `sin^(-1)(f(x))`

Explanation

To solve the problem step by step, we need to find the derivative of the inverse function \( (f^{-1})'(x) \) given that \( f'(x) = \sqrt{1 - (f(x))^2} \).

### Step 1: Understand the relationship between \( f \) and \( f^{-1} \)
Since \( f \) is a bijection, it has an inverse function \( f^{-1} \). By the property of inverse functions, we know that:
\[ f(f^{-1}(x)) = x \]
for all \( x \) in the range of \( f \).
...