Derivative of `sin(sin x^(2))` at `x = sqrt((pi)/(2))` is

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To find the derivative of \( \sin(\sin(x^2)) \) at \( x = \sqrt{\frac{\pi}{2}} \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Differentiate the function
We start with the function:
\[ y = \sin(\sin(x^2)) \]
To differentiate this, we apply the chain rule. The derivative of \( \sin(u) \) is \( \cos(u) \cdot \frac{du}{dx} \).

### Step 2: Apply the chain rule
Let \( u = \sin(x^2) \). Then, we have:
\[ \frac{dy}{dx} = \cos(u) \cdot \frac{du}{dx} \]
Now, we need to find \( \frac{du}{dx} \).

### Step 3: Differentiate \( u = \sin(x^2) \)
Using the chain rule again:
\[ \frac{du}{dx} = \cos(x^2) \cdot \frac{d(x^2)}{dx} = \cos(x^2) \cdot 2x \]

### Step 4: Substitute back into the derivative
Now substituting back into the derivative of \( y \):
\[ \frac{dy}{dx} = \cos(\sin(x^2)) \cdot \cos(x^2) \cdot 2x \]

### Step 5: Evaluate at \( x = \sqrt{\frac{\pi}{2}} \)
Next, we need to evaluate this derivative at \( x = \sqrt{\frac{\pi}{2}} \):
\[ \frac{dy}{dx} \bigg|_{x = \sqrt{\frac{\pi}{2}}} = 2\sqrt{\frac{\pi}{2}} \cdot \cos\left(\sin\left(\left(\sqrt{\frac{\pi}{2}}\right)^2\right)\right) \cdot \cos\left(\left(\sqrt{\frac{\pi}{2}}\right)^2\right) \]
Calculating \( \left(\sqrt{\frac{\pi}{2}}\right)^2 = \frac{\pi}{2} \).

### Step 6: Calculate \( \sin\left(\frac{\pi}{2}\right) \)
Now we find:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus, we have:
\[ \cos\left(\sin\left(\frac{\pi}{2}\right)\right) = \cos(1) \]
And:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Step 7: Substitute values into the derivative
Now substituting these values back:
\[ \frac{dy}{dx} \bigg|_{x = \sqrt{\frac{\pi}{2}}} = 2\sqrt{\frac{\pi}{2}} \cdot \cos(1) \cdot 0 \]
Since one of the multiplicands is 0, we have:
\[ \frac{dy}{dx} \bigg|_{x = \sqrt{\frac{\pi}{2}}} = 0 \]

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