If ` x= sec ^(2) theta, y =tan ^(3) theta ,then " at " theta (pi)/(3) , (dy)/(dx) =`

Options

A. ` (-3sqrt(3))/( 2) `
B. ` (3sqrt(3))/( 2) `
C. ` (-1)/( 2sqrt(3))`
D. ` (1)/( 2sqrt(3))`

Answer

To find \(\frac{dy}{dx}\) when \(x = \sec^2 \theta\) and \(y = \tan^3 \theta\) at \(\theta = \frac{\pi}{3}\), we will follow these steps:

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

Given:
\[ x = \sec^2 \theta \]

Using the differentiation formula for \(\sec^2 \theta\):
\[ \frac{dx}{d\theta} = 2 \sec^2 \theta \tan \theta \]

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

Given:
\[ y = \tan^3 \theta \]

Using the chain rule and the power rule:
\[ \frac{dy}{d\theta} = 3 \tan^2 \theta \cdot \sec^2 \theta \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)

Using the relationship:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{3 \tan^2 \theta \cdot \sec^2 \theta}{2 \sec^2 \theta \tan \theta} \]

### Step 4: Simplify the expression

Cancel \(\sec^2 \theta\) from the numerator and denominator:
\[ \frac{dy}{dx} = \frac{3 \tan^2 \theta}{2 \tan \theta} = \frac{3}{2} \tan \theta \]

### Step 5: Evaluate \(\frac{dy}{dx}\) at \(\theta = \frac{\pi}{3}\)

Now, we substitute \(\theta = \frac{\pi}{3}\):
\[ \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \]

Thus:
\[ \frac{dy}{dx} = \frac{3}{2} \cdot \sqrt{3} = \frac{3\sqrt{3}}{2} \]

### Final Answer:
\[ \frac{dy}{dx} \text{ at } \theta = \frac{\pi}{3} = \frac{3\sqrt{3}}{2} \]

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