If `x=sintheta,y=sin^(3)theta` then `(d^(2)y)/(dx^(2))` at `theta=(pi)/(2)` is . . .

Options

A. 3
B. 6
C. `(1)/(6)`
D. `(1)/(3)`

Explanation

To solve the problem, we need to find the second derivative of \(y\) with respect to \(x\) given the equations \(x = \sin \theta\) and \(y = \sin^3 \theta\) at \(\theta = \frac{\pi}{2}\).

### Step 1: Find \(\frac{dy}{d\theta}\)
We start by differentiating \(y\) with respect to \(\theta\):
\[ y = \sin^3 \theta \]
Using the chain rule:
\[ \frac{dy}{d\theta} = 3 \sin^2 \theta \cdot \frac{d(\sin \theta)}{d\theta} = 3 \sin^2 \theta \cdot \cos \theta \]

### Step 2: Find \(\frac{dx}{d\theta}\)
Next, we differentiate \(x\) with respect to \(\theta\):
\[ x = \sin \theta \]
Thus,
\[ \frac{dx}{d\theta} = \cos \theta \]

### Step 3: Find \(\frac{dy}{dx}\)
Now we can find \(\frac{dy}{dx}\) using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{3 \sin^2 \theta \cos \theta}{\cos \theta} = 3 \sin^2 \theta \]

### Step 4: Find \(\frac{d^2y}{dx^2}\)
To find the second derivative \(\frac{d^2y}{dx^2}\), we differentiate \(\frac{dy}{dx}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(3 \sin^2 \theta\right) \]
Using the chain rule again:
\[ \frac{d^2y}{dx^2} = \frac{d}{d\theta}(3 \sin^2 \theta) \cdot \frac{d\theta}{dx} \]
Calculating \(\frac{d}{d\theta}(3 \sin^2 \theta)\):
\[ \frac{d}{d\theta}(3 \sin^2 \theta) = 6 \sin \theta \cos \theta \]
Now, we need \(\frac{d\theta}{dx}\):
\[ \frac{d\theta}{dx} = \frac{1}{\frac{dx}{d\theta}} = \frac{1}{\cos \theta} \]
Thus,
\[ \frac{d^2y}{dx^2} = 6 \sin \theta \cos \theta \cdot \frac{1}{\cos \theta} = 6 \sin \theta \]

### Step 5: Evaluate at \(\theta = \frac{\pi}{2}\)
Now we substitute \(\theta = \frac{\pi}{2}\):
\[ \frac{d^2y}{dx^2} = 6 \sin\left(\frac{\pi}{2}\right) = 6 \cdot 1 = 6 \]

### Final Answer
Thus, the value of \(\frac{d^2y}{dx^2}\) at \(\theta = \frac{\pi}{2}\) is:
\[ \boxed{6} \]