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
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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; it’s crucial to apply this step properly.
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