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

• A: y = -4x sin(x) cos(x)
• B: y = -4x sin(x) cos(x)
• C: y = 4x sin(x) cos(x)
• D: y = 4x sin(x) cos(x)

To find the derivative of the function \( y = \cos(2x) \), we will use the chain rule. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

Step-by-Step Derivation

1. Identify the outer function and the inner function:

• Outer function: \( \cos(u) \) where \( u = 2x \)
• Inner function: \( u = 2x \)

2. Differentiate the outer function:

The derivative of \( \cos(u) \) is \( -\sin(u) \).

3. Differentiate the inner function:

The derivative of \( u = 2x \) is \( 2 \).

4. Apply the chain rule:

Thus, the derivative \( \frac{dy}{dx} \) is:

\( \frac{dy}{dx} = -\sin(2x) \cdot 2 = -2\sin(2x) \).

Final Result

The derivative of \( y = \cos(2x) \) is:

\( \frac{dy}{dx} = -2\sin(2x) \).

None of the provided options (A, B, C, D) match this result. Therefore, the correct answer is not listed among the choices.