https://file.aiquickdraw.com/mathbot/file/1726513607297zqzn2ugk.jpg

The image you provided contains an algebraic equation and some calculations, which are in Arabic. The main equation and the solution steps are displayed in the image.

### Translation of the Image Content:
The text is stating a condition and calculating a solution:
- Given that \( f(x+1) - f(x) = f'(x) f(x) \), find \( f(x) \).
- Suggests that \( f(x) = ke^{x} \) might be a solution, where \( k \) is a constant.

### Calculation Steps:
1. Substitute \( f(x) = ke^x \) into the given equation:
   \[
   f(x+1) - f(x) = ke^{x+1} - ke^x = ke^x(e - 1)
   \]
2. Then the derivative \( f'(x) \) is:
   \[
   f'(x) = (ke^x)'
   \]
   By applying the derivative, \( f'(x) = ke^x \).
3. Substituting back into the original equation:
   \[
   ke^x(e - 1) = ke^x \cdot ke^x = k^2e^{2x}
   \]
   This leads to:
   \[
   e - 1 = ke^x
   \]
   Here, a contradiction arises, suggesting that the original assumption or some steps might be incorrect.

### Conclusion:
The solution provided in the image ends with a contradiction or error. The function \( f(x) = ke^x \) might not satisfy the original condition \( f(x+1) - f(x) = f'(x) f(x) \) without some additional constraints or corrections in the calculation.

Would you like a more detailed explanation or help with correcting the solution? Here are some related questions you might find helpful:

1. How can the error in the solution be corrected?
2. What conditions on \( k \) might make the function \( f(x) = ke^x \) a valid solution?
3. Can a different function form satisfy the condition \( f(x+1) - f(x) = f'(x) f(x) \)?
4. How does this condition relate to the properties of exponential functions?
5. What are the implications of such functional equations in more complex scenarios?

**Tip:** When working with functional equations involving derivatives, always verify if your assumption about the function's form aligns with all given conditions.

Find f(x) that satisfies f(x+1) - f(x) = 4 f(x) f(x+1) with the condition f(1) = 1, and calculate f(4).

Learn how to solve a functional equation where f(x+1) - f(x) = 4 f(x) f(x+1), given that f(1) = 1. The solution involves algebraic manipulation of functions and is suitable for students in grades 9-12.

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