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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

  1. 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 $u(x) = 2x$.

  2. Differentiate the outer function
    Now we differentiate the outer function with respect to the inner function:
    $$ g'(u) = -2\sin(u) $$

  3. Differentiate the inner function
    Next, we differentiate the inner function with respect to $x$:
    $$ u'(x) = 2 $$

  4. Apply the chain rule
    Using the chain rule, we find the derivative of $f(x)$:
    $$ f'(x) = g'(u) \cdot u'(x) $$

  5. Substitute back the inner function
    Now substitute back $u = 2x$ into the derivative:
    $$ f'(x) = -2\sin(2x) \cdot 2 $$

  6. 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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