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

Options

A. `(-1)/(sqrt(1 - x^(6)))`
B. `(-3x^(2))/(sqrt(1 -x^(6)))`
C. `(-3)/(x^(2) sqrt(1 -x^(6)))`
D. none of these

Explanation

To find the derivative of the function \( y = \cos^{-1}(x^3) \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- The outer function is \( y = \cos^{-1}(u) \) where \( u = x^3 \).
- The inner function is \( u = x^3 \).

2. **Differentiate the outer function**:
- The derivative of \( y = \cos^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{dy}{du} = -\frac{1}{\sqrt{1 - u^2}} \]

3. **Differentiate the inner function**:
- The derivative of \( u = x^3 \) with respect to \( x \) is:
\[ \frac{du}{dx} = 3x^2 \]

4. **Apply the chain rule**:
- According to the chain rule, the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

5. **Substitute the derivatives**:
- Substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the equation:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - (x^3)^2}} \cdot 3x^2 \]

6. **Simplify the expression**:
- This simplifies to:
\[ \frac{dy}{dx} = -\frac{3x^2}{\sqrt{1 - x^6}} \]

### Final Answer:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{3x^2}{\sqrt{1 - x^6}} \]