If ` y= ( cos x ^(2))^(2) , "then" (dy)/(dx) ` is equal to

Options

A. ` - 4x sin 2x^(2)`
B. `- x sin x^(2)`
C. `- 2x sin 2x^(2)`
D. `- x cos 2x^(2)`

Explanation

To find the derivative of the function \( y = (\cos(x^2))^2 \), we will use the chain rule and the product rule of differentiation. Here is the step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = (\cos(x^2))^2 \]

### Step 2: Apply the chain rule
To differentiate \( y \) with respect to \( x \), we will apply the chain rule. The chain rule states that if \( y = u^n \), then:
\[ \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \]
Here, \( u = \cos(x^2) \) and \( n = 2 \).

### Step 3: Differentiate using the chain rule
Differentiating \( y \):
\[ \frac{dy}{dx} = 2(\cos(x^2))^{1} \cdot \frac{d}{dx}(\cos(x^2)) \]

### Step 4: Differentiate \( \cos(x^2) \) using the chain rule
Now, we need to differentiate \( \cos(x^2) \):
\[ \frac{d}{dx}(\cos(x^2)) = -\sin(x^2) \cdot \frac{d}{dx}(x^2) \]
The derivative of \( x^2 \) is \( 2x \), so:
\[ \frac{d}{dx}(\cos(x^2)) = -\sin(x^2) \cdot 2x \]

### Step 5: Substitute back into the derivative of \( y \)
Now, substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 2(\cos(x^2)) \cdot (-\sin(x^2) \cdot 2x) \]
This simplifies to:
\[ \frac{dy}{dx} = -4x \cos(x^2) \sin(x^2) \]

### Step 6: Use the double angle identity
Using the double angle identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \):
\[ \frac{dy}{dx} = -2x \cdot 2 \cos(x^2) \sin(x^2) = -2x \sin(2x^2) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -2x \sin(2x^2) \]

---