We are tasked with finding the derivative of \( y = e^x \tan(x) \) with respect to \( x \). To do this, we'll use the **product rule** and **chain rule**.

The product rule states:
\[
\frac{d}{dx}(u v) = u'v + uv'
\]
where \( u = e^x \) and \( v = \tan(x) \).

### Step 1: Differentiate \( u = e^x \)
\[
u' = \frac{d}{dx}(e^x) = e^x
\]

### Step 2: Differentiate \( v = \tan(x) \)
\[
v' = \frac{d}{dx}(\tan(x)) = \sec^2(x)
\]

### Step 3: Apply the product rule
Using the product rule:
\[
\frac{dy}{dx} = u'v + uv'
\]
Substitute \( u = e^x \), \( u' = e^x \), \( v = \tan(x) \), and \( v' = \sec^2(x) \):
\[
\frac{dy}{dx} = e^x \tan(x) + e^x \sec^2(x)
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = e^x (\tan(x) + \sec^2(x))
\]

Would you like further details on any part of the solution?

Here are five related questions:
1. How do you apply the chain rule in differentiation?
2. What is the derivative of \( e^{kx} \) where \( k \) is a constant?
3. Can you differentiate \( \tan(x) \) using the quotient rule?
4. How do you differentiate a product of three functions?
5. How do you integrate \( \sec^2(x) \)?

**Tip**: The derivative of a product involves differentiating each part separately and then combining the results.

Explore how to differentiate the function y = e^x tan(x) step by step using the product rule and chain rule. This solution explains key calculus concepts and demonstrates the application of the product rule for differentiation. Suitable for advanced high school or undergraduate students.

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