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To differentiate the given expression:

\[
f(x) = \frac{1 + \sin(2x)}{\cos(2x)}
\]

we'll apply the quotient rule for derivatives. The quotient rule states:

\[
\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]

Here, \( u(x) = 1 + \sin(2x) \) and \( v(x) = \cos(2x) \).

### Step 1: Differentiate \( u(x) \)
\[
u'(x) = \frac{d}{dx}(1 + \sin(2x)) = 0 + 2\cos(2x) = 2\cos(2x)
\]

### Step 2: Differentiate \( v(x) \)
\[
v'(x) = \frac{d}{dx}(\cos(2x)) = -2\sin(2x)
\]

### Step 3: Apply the quotient rule
Now, applying the quotient rule:
\[
f'(x) = \frac{\cos(2x) \cdot 2\cos(2x) - (1 + \sin(2x)) \cdot (-2\sin(2x))}{\cos^2(2x)}
\]

### Step 4: Simplify the expression
Simplify the numerator:
\[
f'(x) = \frac{2\cos^2(2x) + 2(1 + \sin(2x))\sin(2x)}{\cos^2(2x)}
\]

Further simplification:
\[
f'(x) = \frac{2\cos^2(2x) + 2\sin(2x) + 2\sin^2(2x)}{\cos^2(2x)}
\]

This is the derivative of the given function. You can simplify further based on trigonometric identities if necessary.

Would you like more details on the simplification or any other part of the process?

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Here are some follow-up questions:
1. How would you use the chain rule to differentiate \( \sin(2x) \)?
2. What are the conditions for the quotient rule to be valid?
3. Can the result be simplified using a specific trigonometric identity?
4. What happens to the derivative as \( x \) approaches 0?
5. How does the second derivative of this function look?

**Tip:** When using the quotient rule, always check if simplification before differentiation can make the process easier.

Differentiate the expression: (1 + sin(2x)) / cos(2x)

Learn how to differentiate the expression (1 + sin(2x)) / cos(2x) using the quotient rule. This step-by-step solution covers key calculus concepts like the quotient rule and chain rule, along with trigonometric differentiation. Suitable for undergraduate calculus students.

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