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We are tasked with finding the function whose derivative is given as \( f'(x) = e^{4x} \) and which passes through the point \( P(0, \frac{9}{4}) \).

### Steps:

1. **Find the general form of the function \( f(x) \):**
   To find \( f(x) \), we need to integrate \( f'(x) = e^{4x} \).

   \[
   f(x) = \int e^{4x} \, dx
   \]

   Using basic integration rules, we have:

   \[
   f(x) = \frac{1}{4} e^{4x} + C
   \]

   where \( C \) is the constant of integration.

2. **Use the point \( P(0, \frac{9}{4}) \) to find \( C \):**
   We are given that the function passes through the point \( P(0, \frac{9}{4}) \). Substituting \( x = 0 \) and \( f(0) = \frac{9}{4} \) into the equation:

   \[
   f(0) = \frac{1}{4} e^{4(0)} + C = \frac{9}{4}
   \]
   
   Simplifying:

   \[
   \frac{1}{4} \cdot 1 + C = \frac{9}{4}
   \]

   \[
   \frac{1}{4} + C = \frac{9}{4}
   \]

   Subtract \( \frac{1}{4} \) from both sides:

   \[
   C = \frac{9}{4} - \frac{1}{4} = \frac{8}{4} = 2
   \]

3. **Final function:**
   Now that we know \( C = 2 \), the final function is:

   \[
   f(x) = \frac{1}{4} e^{4x} + 2
   \]

### Final Answer:
The function is \( f(x) = \frac{1}{4} e^{4x} + 2 \).

---

Would you like more details or have any questions about the solution?

Here are 5 related questions:
1. How would the function change if the derivative was \( f'(x) = e^{5x} \)?
2. What happens to the constant of integration if the point \( P \) was different?
3. Can you explain the role of the constant \( C \) in the general solution?
4. What are some practical applications of solving such differential equations?
5. How can we confirm that the function satisfies the given derivative and point?

**Tip:** Always remember that when integrating, the constant of integration \( C \) must be determined by additional information, such as a given point.

Find the function with the derivative f'(x) = e^(4x) whose graph passes through the point P(0, 9/4).

This problem demonstrates how to find a function given its derivative f'(x) = e^(4x) and the condition that it passes through the point P(0, 9/4). It involves key concepts from calculus, including integration and solving for constants.

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