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.