If y = cos² x², find dy / dx?

Given:

• y = cos²(x²)

Concept Used:

• The chain rule is used for differentiation:

  • If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).

• For cos²(u), where u is a function of x:

  • d(cos²(u))/du = -2 × cos(u) × sin(u).

Calculation:

Step 1: Differentiate y = cos²(x²) using the chain rule:

⇒ dy/dx = d(cos²(x²))/d(x²) × d(x²)/dx

Step 2: Differentiate cos²(x²) with respect to x²:

⇒ d(cos²(x²))/d(x²) = -2 × cos(x²) × sin(x²)

Step 3: Differentiate x² with respect to x:

⇒ d(x²)/dx = 2x

Step 4: Combine the results:

⇒ dy/dx = (-2 × cos(x²) × sin(x²)) × (2x)

⇒ dy/dx = -4x × cos(x²) × sin(x²)

Conclusion:

∴ dy/dx = -4x × cos(x²) × sin(x²)

The correct answer is: Option 2.