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.
Answer
-2\sin(2x)
Answer for screen readers
The final answer is -2sin(2x)
Steps to Solve
- 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) ).
$$ \frac{d}{du}[\cos(u)] = -\sin(u) $$
- Differentiate the inner function
The inner function is ( 2x ). The derivative of ( 2x ) with respect to ( x ) is simply 2.
$$ \frac{d}{dx}[2x] = 2 $$
- 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.
$$ \frac{d}{dx}[\cos(2x)] = -\sin(2x) \cdot 2 $$
- Simplify the expression
Combine the terms to get the final derivative.
$$ \frac{d}{dx}[\cos(2x)] = -2\sin(2x) $$
The final answer is -2sin(2x)
More Information
Using the chain rule in calculus is essential for finding derivatives of composite functions.
Tips
Students often forget to apply the chain rule, especially remembering to multiply by the derivative of the inner function. Always identify the inner and outer functions clearly to avoid this mistake.
