Step-by-step Solution:

Given: \[ y = \cos^2(x^2) \] Step 1: Apply the chain rule Using the chain rule, differentiate \( y = \cos^2(u) \) with respect to \( u \), and \( u = x^2 \): \[ \frac{dy}{dx} = \frac{d}{du}(\cos^2(u)) \cdot \frac{du}{dx} \] Step 2: Differentiate \( \cos^2(u) \) with respect to \( u \) The derivative of \( \cos^2(u) \) is: \[ \frac{d}{du}(\cos^2(u)) = 2\cos(u)(-\sin(u)) = -2\cos(u)\sin(u) \] Step 3: Differentiate \( u = x^2 \) with respect to \( x \) \[ \frac{du}{dx} = 2x \] Step 4: Substitute back \( u = x^2 \) \[ \frac{dy}{dx} = -2\cos(u)\sin(u) \cdot 2x \] \[ \frac{dy}{dx} = -4x\cos(x^2)\sin(x^2) \]