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To find the function \( f(x) \) with the derivative \( f'(x) = e^{5x} \) that passes through the point \( P = (0, \frac{8}{5}) \), let's follow these steps:

1. **Integrate \( f'(x) = e^{5x} \) to find \( f(x) \):**
   \[
   f(x) = \int e^{5x} \, dx
   \]
   To integrate \( e^{5x} \), use the formula \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \).

   Thus,
   \[
   f(x) = \frac{1}{5} e^{5x} + C
   \]
   where \( C \) is the constant of integration.

2. **Use the point \( P = (0, \frac{8}{5}) \) to solve for \( C \):**

   Substitute \( x = 0 \) and \( f(0) = \frac{8}{5} \) into the equation:
   \[
   f(0) = \frac{1}{5} e^{5 \cdot 0} + C = \frac{1}{5} \cdot 1 + C = \frac{1}{5} + C
   \]
   Since \( f(0) = \frac{8}{5} \), we have:
   \[
   \frac{1}{5} + C = \frac{8}{5}
   \]
   Solving for \( C \):
   \[
   C = \frac{8}{5} - \frac{1}{5} = \frac{7}{5}
   \]

3. **Write the final function:**

   Substitute \( C = \frac{7}{5} \) back into the expression for \( f(x) \):
   \[
   f(x) = \frac{1}{5} e^{5x} + \frac{7}{5}
   \]

### Answer:
\[
f(x) = \frac{1}{5} e^{5x} + \frac{7}{5}
\]

Would you like more details or have any questions?

### Related Questions
1. How would the solution change if \( f(x) \) passed through a different point?
2. What is the significance of the constant \( C \) in the integration process?
3. How does the value of the coefficient in the exponent affect the integration result?
4. What is the general approach to solving for constants in integration when given a specific point?
5. How can you verify that \( f(x) \) satisfies the derivative \( f'(x) = e^{5x} \) and passes through \( P \)?

### Tip:
When finding an antiderivative with an initial condition, always use the given point to solve for the constant after integrating.

Find the function with derivative f'(x) = e^(5x) that passes through the point P = (0, 8/5).

Learn how to find the function f(x) with derivative e^(5x) that passes through the point (0, 8/5). This solution involves calculus concepts such as antiderivatives, exponential functions, and integration. Suitable for high school calculus students in Grades 11-12.

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