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Let's solve both questions by differentiating each function with respect to \( x \).

### Question 12
The function given is:
\[
y = e^x \cdot 3^x
\]
To differentiate \( y \) with respect to \( x \), we use the **product rule**:
\[
\frac{d}{dx}(u \cdot v) = u'v + uv'
\]
where \( u = e^x \) and \( v = 3^x \).

1. **Differentiate \( u = e^x \):**
   \[
   u' = e^x
   \]

2. **Differentiate \( v = 3^x \):**
   Since \( 3^x = e^{x \ln 3} \), we can differentiate it as:
   \[
   v' = 3^x \ln 3
   \]

Now, apply the product rule:
\[
y' = e^x \cdot 3^x \ln 3 + e^x \cdot 3^x
\]
Factor out \( e^x \cdot 3^x \):
\[
y' = e^x \cdot 3^x (\ln 3 + 1)
\]

### Question 13
The function given is:
\[
y = e^x \cdot \ln(1 + x^2)
\]
Again, we use the **product rule** where \( u = e^x \) and \( v = \ln(1 + x^2) \).

1. **Differentiate \( u = e^x \):**
   \[
   u' = e^x
   \]

2. **Differentiate \( v = \ln(1 + x^2) \):**
   Using the chain rule:
   \[
   v' = \frac{1}{1 + x^2} \cdot 2x = \frac{2x}{1 + x^2}
   \]

Now, apply the product rule:
\[
y' = e^x \cdot \ln(1 + x^2) + e^x \cdot \frac{2x}{1 + x^2}
\]
Factor out \( e^x \):
\[
y' = e^x \left( \ln(1 + x^2) + \frac{2x}{1 + x^2} \right)
\]

### Summary of Answers
1. **For Question 12:**
   \[
   y' = e^x \cdot 3^x (\ln 3 + 1)
   \]

2. **For Question 13:**
   \[
   y' = e^x \left( \ln(1 + x^2) + \frac{2x}{1 + x^2} \right)
   \]

Would you like further details on these steps?

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Here are some additional related questions to consider:

1. How would you differentiate a function with a product of three terms?
2. What is the derivative of \( 3^x \) in terms of its base and exponent?
3. How does the chain rule apply to logarithmic functions?
4. Can we simplify the expression for \( y' \) in Question 13 further?
5. How does the natural logarithm's derivative affect complex functions?

**Tip:** For exponential and logarithmic functions, always remember to apply the chain rule when there’s a composite function inside.

Differentiate the following functions with respect to x: (1) y = e^x * 3^x and (2) y = e^x * log(1 + x^2)

Learn how to differentiate functions involving exponential terms and logarithms, such as y = e^x * 3^x and y = e^x * log(1 + x^2). This solution covers the product rule, chain rule, and logarithmic differentiation.

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