Find the derivative of `cos( sin x^(2)) at x = sqrt(pi/2)`

Explanation

To find the derivative of the function \( f(x) = \cos(\sin(x^2)) \) at \( x = \sqrt{\frac{\pi}{2}} \), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ f(x) = \cos(\sin(x^2)) \]
To find the derivative \( f'(x) \), we will use the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \cdot \frac{du}{dx} \), where \( u = \sin(x^2) \).

First, we differentiate \( \cos(u) \):
\[ f'(x) = -\sin(\sin(x^2)) \cdot \frac{d}{dx}(\sin(x^2)) \]

### Step 2: Differentiate \( \sin(x^2) \)
Next, we need to differentiate \( \sin(x^2) \) using the chain rule again:
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

### Step 3: Combine the derivatives
Now we can substitute back into our expression for \( f'(x) \):
\[ f'(x) = -\sin(\sin(x^2)) \cdot (2x \cos(x^2)) \]
Thus, we have:
\[ f'(x) = -2x \cos(x^2) \sin(\sin(x^2)) \]

### Step 4: Evaluate the derivative at \( x = \sqrt{\frac{\pi}{2}} \)
Now we need to evaluate \( f'(\sqrt{\frac{\pi}{2}}) \):
\[ f'(\sqrt{\frac{\pi}{2}}) = -2\left(\sqrt{\frac{\pi}{2}}\right) \cos\left(\left(\sqrt{\frac{\pi}{2}}\right)^2\right) \sin\left(\sin\left(\left(\sqrt{\frac{\pi}{2}}\right)^2\right)\right) \]
Calculating \( \left(\sqrt{\frac{\pi}{2}}\right)^2 \):
\[ \left(\sqrt{\frac{\pi}{2}}\right)^2 = \frac{\pi}{2} \]
So we have:
\[ f'(\sqrt{\frac{\pi}{2}}) = -2\left(\sqrt{\frac{\pi}{2}}\right) \cos\left(\frac{\pi}{2}\right) \sin\left(\sin\left(\frac{\pi}{2}\right)\right) \]

### Step 5: Evaluate \( \cos\left(\frac{\pi}{2}\right) \) and \( \sin\left(\frac{\pi}{2}\right) \)
We know:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus, substituting these values:
\[ f'(\sqrt{\frac{\pi}{2}}) = -2\left(\sqrt{\frac{\pi}{2}}\right) \cdot 0 \cdot \sin(1) \]
This simplifies to:
\[ f'(\sqrt{\frac{\pi}{2}}) = 0 \]

### Final Answer
The derivative of \( \cos(\sin(x^2)) \) at \( x = \sqrt{\frac{\pi}{2}} \) is:
\[ \boxed{0} \]