Step 1: Understanding the Question:
The question requires us to find the derivative of \(y\) with respect to \(x\), where \(y\) is a composite function involving the inverse sine function.
Step 2: Key Formula or Approach:
The problem can be solved using the chain rule for differentiation. The formula for the derivative of the inverse sine function is:
\[
\frac{d}{dx}\left(\sin^{-1}(u)\right) = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}
\]
where \(u\) is a function of \(x\).
Step 3: Detailed Explanation:
In this problem, the function is \( y = \sin^{-1}(3x - 4x^3) \). Let's identify the inner function \(u\):
\[
u = 3x - 4x^3
\]
First, find the derivative of \(u\) with respect to \(x\):
\[
\frac{du}{dx} = \frac{d}{dx}(3x - 4x^3) = 3 - 12x^2
\]
Now, apply the chain rule formula:
\[
\frac{dy}{dx} = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}
\]
Substitute the expressions for \(u\) and \(\frac{du}{dx}\):
\[
\frac{dy}{dx} = \frac{1}{\sqrt{1-(3x-4x^3)^2}} \cdot (3-12x^2)
\]
Step 4: Final Answer:
Combining the terms into a single fraction gives the final derivative in the required form:
\[
\frac{dy}{dx} = \frac{3-12x^2}{\sqrt{1-(3x-4x^3)^2}}
\]
This matches option (A).