If `y=cos^(2)x^(2)`, find `(dy)/(dx)`.

Options

A. `y=-4xsinx^(2)cosx^(2)`
B. `y=-4xsinx^(3)cosx^(2)`
C. `y=4xsinx^(2)cosx^(2)`
D. `y=4xsinx^(3)cosx^(2)`

Explanation

To find the derivative of the function \( y = \cos^2(x^2) \) with respect to \( x \), we will use the chain rule and the power rule of differentiation. Here’s the step-by-step solution:

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

### Step 2: Apply the chain rule
To differentiate \( y \), we will first differentiate the outer function \( u^2 \) where \( u = \cos(x^2) \). The derivative of \( u^2 \) is:
\[ \frac{dy}{du} = 2u \]

### Step 3: Differentiate the inner function
Next, we need to differentiate the inner function \( u = \cos(x^2) \). Using the chain rule again:
\[ \frac{du}{dx} = -\sin(x^2) \cdot \frac{d}{dx}(x^2) \]
The derivative of \( x^2 \) is \( 2x \), so:
\[ \frac{du}{dx} = -\sin(x^2) \cdot 2x \]

### Step 4: Combine the derivatives
Now, we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \frac{du}{dx} \]
Substituting \( u = \cos(x^2) \) and \( \frac{du}{dx} = -\sin(x^2) \cdot 2x \):
\[ \frac{dy}{dx} = 2\cos(x^2) \cdot (-\sin(x^2) \cdot 2x) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -4x \cos(x^2) \sin(x^2) \]

### Final Answer
Thus, the derivative of \( y = \cos^2(x^2) \) is:
\[ \frac{dy}{dx} = -4x \cos(x^2) \sin(x^2) \]

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