You are given the functions f(x) and g(x) = sin^-1(fx). Which of the following expressions represents the ratio of the derivatives of these two functions, f'(x)/g'(x)? Assume that the domain of g(x) is -pi/2 < f(x) < pi/2. A. f'(x)/g'(x) = 1/(sqrt( 1 - (f(x))^2 )
B. f'(x)/g'(x) = 1/(sqrt( 1 + f(x) )
C. f'(x)/g'(x) = (sqrt( 1 - (f(x))^2 )
D. f'(x)/g'(x) = (sqrt( 1 - f(x) )
E. f'(x)/g'(x) = (sqrt( 1 + f^2(x) )

C

Step-by-step explanation:

We are given the two functions:

And we want to find the ratio that represents:

To do so, we will compute the derivatives of both functions.

For f,  we can simply write that:

For g, we will use the chain rule:

We can let:

We can double check this by doing the composition:

Therefore, by the chain rule, we acquire that:

By substitution:

Hence, our ratio is:

Simplify:

Hence, our answer is C