# differentiate cos(2x)

#### Understand the Problem

The question is asking for the mathematical differentiation of the function cos(2x). This involves applying the rules of differentiation, particularly the chain rule, to find the derivative of this trigonometric function.

-2\sin(2x)

#### Steps to Solve

1. Differentiate the outer function

The outer function is ( \cos(u) ), where ( u = 2x ). The derivative of ( \cos(u) ) with respect to ( u ) is ( -\sin(u) ).

$$\frac{d}{du}[\cos(u)] = -\sin(u)$$

1. Differentiate the inner function

The inner function is ( 2x ). The derivative of ( 2x ) with respect to ( x ) is simply 2.

$$\frac{d}{dx}[2x] = 2$$

1. Apply the chain rule

According to the chain rule, the derivative of ( \cos(2x) ) is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

$$\frac{d}{dx}[\cos(2x)] = -\sin(2x) \cdot 2$$

1. Simplify the expression

Combine the terms to get the final derivative.

$$\frac{d}{dx}[\cos(2x)] = -2\sin(2x)$$