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The problem asks us to find the derivative of the function \( f(x) = 3x^3 (4^x) \) at \( x = 1 \).

### Step-by-Step Solution:

1. **Identify the Function and Apply Product Rule:**
   The function \( f(x) = 3x^3 \cdot 4^x \) is a product of two functions:
   - \( u(x) = 3x^3 \)
   - \( v(x) = 4^x \)
   
   We can use the product rule for differentiation:
   \[
   f'(x) = u'(x)v(x) + u(x)v'(x)
   \]

2. **Differentiate Each Part:**
   - For \( u(x) = 3x^3 \):
     \[
     u'(x) = 3 \cdot 3x^2 = 9x^2
     \]
   - For \( v(x) = 4^x \):
     \[
     v'(x) = 4^x \ln(4)
     \]
     (since the derivative of \( a^x \) is \( a^x \ln(a) \)).

3. **Apply the Product Rule:**
   Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \) into the product rule formula:
   \[
   f'(x) = (9x^2)(4^x) + (3x^3)(4^x \ln(4))
   \]
   Simplify by factoring out \( 4^x \):
   \[
   f'(x) = 4^x (9x^2 + 3x^3 \ln(4))
   \]

4. **Evaluate at \( x = 1 \):**
   Substitute \( x = 1 \) into \( f'(x) \):
   \[
   f'(1) = 4^1 (9 \cdot 1^2 + 3 \cdot 1^3 \ln(4))
   \]
   Simplify further:
   \[
   f'(1) = 4 (9 + 3 \ln(4))
   \]
   \[
   f'(1) = 36 + 12 \ln(4)
   \]

### Answer:
The correct answer is:
\[
\boxed{\text{a. } f'(1) = 12 \ln 4 + 36}
\]

Would you like further details on the steps, or have any other questions?

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### Related Questions
1. How would you apply the product rule if the function had three terms instead of two?
2. What is the derivative of \( 4^x \) with respect to \( x \)?
3. How would the answer change if \( f(x) = 3x^3 e^x \) instead?
4. Why is the natural logarithm used in the derivative of an exponential function?
5. How do you evaluate \( f'(x) \) if given \( x = 2 \) instead?

### Tip
Always look for common factors in derivatives to simplify expressions effectively, especially when applying the product rule.

Given the function f(x) = 3x^3 (4^x), find the derivative at x = 1.

Calculate the derivative of f(x) = 3x^3 (4^x) at x = 1 using the product rule and logarithmic differentiation. This solution covers differentiation techniques for exponential functions and the product rule. Suitable for Grades 11-12.

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