If `y=cos(sinx^2)`,then at `x=sqrt((pi)/(2))`,`(dy)/(dx)=`

Options

A. -2
B. 2
C. `-2sqrt((pi)/(2))`
D. 0

Explanation

To solve the problem, we need to find the derivative of \( y = \cos(\sin(x^2)) \) with respect to \( x \) and evaluate it at \( x = \sqrt{\frac{\pi}{2}} \).

### Step-by-Step Solution:

1. **Identify the function**: We have \( y = \cos(\sin(x^2)) \).

2. **Differentiate using the chain rule**:
\[ \frac{dy}{dx} = \frac{d}{dx}[\cos(\sin(x^2))] = -\sin(\sin(x^2)) \cdot \frac{d}{dx}[\sin(x^2)] \]

3. **Differentiate \( \sin(x^2) \)**:
\[ \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}[x^2] = \cos(x^2) \cdot 2x \]

4. **Substituting back**: Now substitute this back into the derivative:
\[ \frac{dy}{dx} = -\sin(\sin(x^2)) \cdot (\cos(x^2) \cdot 2x) \]

5. **Evaluate at \( x = \sqrt{\frac{\pi}{2}} \)**:
- First, calculate \( x^2 \):
\[ x^2 = \left(\sqrt{\frac{\pi}{2}}\right)^2 = \frac{\pi}{2} \]
- Now substitute \( x^2 \) into the expression:
\[ \frac{dy}{dx} = -\sin(\sin(\frac{\pi}{2})) \cdot (\cos(\frac{\pi}{2}) \cdot 2\sqrt{\frac{\pi}{2}}) \]

6. **Evaluate \( \sin(\frac{\pi}{2}) \) and \( \cos(\frac{\pi}{2}) \)**:
- \( \sin(\frac{\pi}{2}) = 1 \)
- \( \cos(\frac{\pi}{2}) = 0 \)

7. **Final evaluation**:
\[ \frac{dy}{dx} = -\sin(1) \cdot (0 \cdot 2\sqrt{\frac{\pi}{2}}) = 0 \]

Thus, the final answer is:
\[ \frac{dy}{dx} \bigg|_{x = \sqrt{\frac{\pi}{2}}} = 0 \]