If ` x=cos ^(-1) (4t^(3) -3t) ,y =tan ^(-1)((sqrt( 1-t^(2)))/( t)),then (dy)/(dx)=`

Options

A. ` (1)/(3) `
B. ` (-1)/(3)`
C. ` 3`
D. `-3`

Answer

To solve the problem, we need to find \(\frac{dy}{dx}\) given the parametric equations:

1. \( x = \cos^{-1}(4t^3 - 3t) \)
2. \( y = \tan^{-1}\left(\frac{\sqrt{1 - t^2}}{t}\right) \)

### Step 1: Differentiate \(x\) with respect to \(t\)

To differentiate \(x\), we will use the chain rule. First, we need to differentiate the inverse cosine function:

\[ \frac{dx}{dt} = -\frac{1}{\sqrt{1 - (4t^3 - 3t)^2}} \cdot \frac{d}{dt}(4t^3 - 3t) \]

Now, we differentiate \(4t^3 - 3t\):

\[ \frac{d}{dt}(4t^3 - 3t) = 12t^2 - 3 \]

Thus, we have:

\[ \frac{dx}{dt} = -\frac{12t^2 - 3}{\sqrt{1 - (4t^3 - 3t)^2}} \]

### Step 2: Differentiate \(y\) with respect to \(t\)

Now we differentiate \(y\):

\[ \frac{dy}{dt} = \frac{1}{1 + \left(\frac{\sqrt{1 - t^2}}{t}\right)^2} \cdot \frac{d}{dt}\left(\frac{\sqrt{1 - t^2}}{t}\right) \]

First, we simplify \(\left(\frac{\sqrt{1 - t^2}}{t}\right)^2\):

\[ \left(\frac{\sqrt{1 - t^2}}{t}\right)^2 = \frac{1 - t^2}{t^2} \]

Thus, we have:

\[ 1 + \left(\frac{\sqrt{1 - t^2}}{t}\right)^2 = \frac{1}{t^2} \]

Now, we differentiate \(\frac{\sqrt{1 - t^2}}{t}\) using the quotient rule:

\[ \frac{d}{dt}\left(\frac{\sqrt{1 - t^2}}{t}\right) = \frac{t \cdot \frac{-t}{\sqrt{1 - t^2}} - \sqrt{1 - t^2}}{t^2} \]

This simplifies to:

\[ \frac{-t^2 - (1 - t^2)}{t^2 \sqrt{1 - t^2}} = \frac{-1}{t^2 \sqrt{1 - t^2}} \]

Thus, we have:

\[ \frac{dy}{dt} = \frac{1}{1 + \frac{1 - t^2}{t^2}} \cdot \frac{-1}{t^2 \sqrt{1 - t^2}} = \frac{t^2}{1}{\cdot} \frac{-1}{t^2 \sqrt{1 - t^2}} = \frac{-1}{\sqrt{1 - t^2}} \]

### Step 3: Find \(\frac{dy}{dx}\)

Now we can find \(\frac{dy}{dx}\) using the chain rule:

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-\frac{1}{\sqrt{1 - t^2}}}{-\frac{12t^2 - 3}{\sqrt{1 - (4t^3 - 3t)^2}}} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - t^2}} \cdot \frac{\sqrt{1 - (4t^3 - 3t)^2}}{12t^2 - 3} \]

### Step 4: Evaluate the expression

Now, we need to evaluate this expression to find the final result.

After simplification, we find:

\[ \frac{dy}{dx} = \frac{1}{3} \]

### Final Answer

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

\[ \frac{dy}{dx} = \frac{1}{3} \]