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

Options

A. `(3)/(x^(4)*sin[cos^(-1((1/(x^(3))))]`
B. `(3)/(x^(2)*sin[cos^(-1)(1)/(x^(3)))]`
C. `(3)/(xsqrt(x^(6)-1))`
D. `(-3)/(xsqrt(x^(6)-1))`

Answer

To find the derivative of the function \( y = \cos^{-1}\left(\frac{1}{x^3}\right) \), we will follow these steps:

### Step 1: Differentiate using the derivative formula for \( \cos^{-1}(x) \)

The derivative of \( \cos^{-1}(u) \) with respect to \( x \) is given by:

\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]

where \( u = \frac{1}{x^3} \).

### Step 2: Find \( \frac{du}{dx} \)

Now, we need to differentiate \( u = \frac{1}{x^3} \):

\[ u = x^{-3} \]

Using the power rule:

\[ \frac{du}{dx} = -3x^{-4} = -\frac{3}{x^4} \]

### Step 3: Substitute \( u \) into the derivative formula

Next, we substitute \( u \) back into the derivative formula. We first need to calculate \( 1 - u^2 \):

\[ u^2 = \left(\frac{1}{x^3}\right)^2 = \frac{1}{x^6} \]

Thus,

\[ 1 - u^2 = 1 - \frac{1}{x^6} = \frac{x^6 - 1}{x^6} \]

### Step 4: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula

Now we can substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula:

\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{1}{x^3}\right)^2}} \cdot \left(-\frac{3}{x^4}\right) \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{3}{x^4 \sqrt{1 - \frac{1}{x^6}}} \]

### Step 5: Simplify the expression

Now we simplify the square root:

\[ \sqrt{1 - \frac{1}{x^6}} = \sqrt{\frac{x^6 - 1}{x^6}} = \frac{\sqrt{x^6 - 1}}{x^3} \]

Substituting this back into our derivative gives:

\[ \frac{dy}{dx} = \frac{3}{x^4} \cdot \frac{x^3}{\sqrt{x^6 - 1}} = \frac{3x^3}{x^4 \sqrt{x^6 - 1}} = \frac{3}{x \sqrt{x^6 - 1}} \]

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{3}{x \sqrt{x^6 - 1}} \]

---