If `y = sin^(-1) (3x -4x^(3)) " then " (dy)/(dx) =` ?

Options

A. `(3)/(sqrt(1 -x^(2)))`
B. `(-4)/(sqrt(1 -x^(2)))`
C. `(3)/(sqrt(1 + x^(3)))`
D. none of these

Explanation

To find the derivative of \( y = \sin^{-1}(3x - 4x^3) \), we will follow these steps:

### Step 1: Differentiate using the chain rule
We know that the derivative of \( \sin^{-1}(u) \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = 3x - 4x^3 \).

### Step 2: Find \( \frac{du}{dx} \)
Now, we need to compute \( \frac{du}{dx} \):
\[ u = 3x - 4x^3 \]
Differentiating \( u \) with respect to \( x \):
\[ \frac{du}{dx} = 3 - 12x^2 \]

### Step 3: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula
Now we substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (3x - 4x^3)^2}} \cdot (3 - 12x^2) \]

### Step 4: Simplify the expression
The expression for \( \frac{dy}{dx} \) becomes:
\[ \frac{dy}{dx} = \frac{3 - 12x^2}{\sqrt{1 - (3x - 4x^3)^2}} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{3 - 12x^2}{\sqrt{1 - (3x - 4x^3)^2}} \]

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