Answer:
C
Step-by-step explanation:
We are given the two functions:
[tex]\displaystyle f(x) \text{ and } g(x)=\sin^{-1}(f(x))[/tex]
And we want to find the ratio that represents:
[tex]\displaystyle \frac{f^\prime(x)}{g^\prime(x)}[/tex]
To do so, we will compute the derivatives of both functions.
For f, we can simply write that:
[tex]f^\prime(x)=f^\prime(x)[/tex]
For g, we will use the chain rule:
[tex]\displaystyle (u(v(x))^\prime=u^\prime(v(x))\cdot v'(x)[/tex]
We can let:
[tex]u(x)=\sin^{-1}(x)\text{ and } v(x)=f(x)[/tex]
We can double check this by doing the composition:
[tex]g(x)=u(v(x))=\sin^{-1}(f(x))[/tex]
Therefore, by the chain rule, we acquire that:
[tex]\displaystyle g^\prime(x)=\frac{1}{\sqrt{1-(v(x))^2}}\cdot v'(x)[/tex]
By substitution:
[tex]\displaystyle g^\prime(x)=\frac{f^\prime(x)}{\sqrt{1-(f(x))^2}}[/tex]
Hence, our ratio is:
[tex]\displaystyle \frac{f^\prime(x)}{g^\prime(x)}=\frac{f^\prime(x)}{\dfrac{f^\prime(x)}{\sqrt{1-(f(x))^2}}}[/tex]
Simplify:
[tex]\displaystyle \frac{f^\prime(x)}{g^\prime(x)}=\sqrt{1-(f(x))^2}[/tex]
Hence, our answer is C
