derivative of 2cos(2x)
Understand the Problem
The question is asking for the derivative of the function 2cos(2x). We will apply the chain rule to differentiate this function, as it is a cosine function multiplied by a constant.
Answer
The derivative is $4\sin(2x)$.
Answer for screen readers
The derivative of the function $2\cos(2x)$ is $4\sin(2x)$.
Steps to Solve

Identify the outer and inner functions
In the function $f(x) = 2\cos(2x)$, the outer function is $g(u) = 2\cos(u)$ and the inner function is $u(x) = 2x$. 
Differentiate the outer function
Now we differentiate the outer function with respect to the inner function:
$$ g'(u) = 2\sin(u) $$ 
Differentiate the inner function
Next, we differentiate the inner function with respect to $x$:
$$ u'(x) = 2 $$ 
Apply the chain rule
Using the chain rule, we find the derivative of $f(x)$:
$$ f'(x) = g'(u) \cdot u'(x) $$ 
Substitute back the inner function
Now substitute back $u = 2x$ into the derivative:
$$ f'(x) = 2\sin(2x) \cdot 2 $$ 
Simplify the final expression
Multiply to simplify:
$$ f'(x) = 4\sin(2x) $$
The derivative of the function $2\cos(2x)$ is $4\sin(2x)$.
More Information
This derivative shows the rate of change of the function $2\cos(2x)$ at any point $x$. It's essential in physics and engineering for understanding wave behavior, among other things.
Tips
 Forgetting to apply the chain rule correctly. Remember to differentiate both the outer and inner functions.
 Neglecting to multiply by the derivative of the inner function; itâ€™s crucial to apply this step properly.