derivative of sin(2x) cos(2x)

Steps to Solve

1. Identify the function and its components

The function we need to differentiate is $f(x) = \sin(2x) \cdot \cos(2x)$. Let's identify the two parts of the product:

Let $u = \sin(2x)$ and $v = \cos(2x)$.

2. Apply the product rule

The product rule states that if you have a product of two functions, the derivative is given by:

$$ \frac{d}{dx}(uv) = u'v + uv' $$

Therefore, we will calculate $u'$ and $v'$ first.

3. Differentiate u and v

Now, we differentiate $u$ and $v$:

For $u = \sin(2x)$, using the chain rule, we have:

$$ u' = 2\cos(2x) $$

For $v = \cos(2x)$, using the chain rule, we have:

$$ v' = -2\sin(2x) $$

4. Substitute u, v, u', and v' into the product rule

Now, we can substitute these values into the product rule formula:

$$ f'(x) = u'v + uv' $$

Replacing $u$, $v$, $u'$, and $v'$ gives:

$$ f'(x) = (2\cos(2x)) \cdot \cos(2x) + \sin(2x) \cdot (-2\sin(2x)) $$

5. Simplify the expression

Now let's simplify the expression:

$$ f'(x) = 2\cos^2(2x) - 2\sin^2(2x) $$

This can be factored using the identity $2\cos^2(a) - 2\sin^2(a) = 2\cos(2a)$:

$$ f'(x) = 2(\cos^2(2x) - \sin^2(2x)) = 2\cos(4x) $$