To find a function $f(x)$ such that its derivative equals the function composed with itself, i.e.,

$f'(x) = f(f(x)),$

we'll try an approach of hypothesizing possible functional forms. This is a relatively complex differential equation, but let’s break it down.

Step 1: Assume a simple form for $f(x)$

Let’s start with the assumption that $f(x)$ might be a simple function like $f(x) = e^x$.

Step 2: Check $f(x) = e^x$

For $f(x) = e^x$, we compute the derivative and the composition:

• $f'(x) = e^x$.
• $f(f(x)) = e^{e^x}$.

Clearly, $f'(x) \neq f(f(x))$ in this case.

Step 3: Try a different hypothesis: $f(x) = e^{g(x)}$

Now, let’s hypothesize a form $f(x) = e^{g(x)}$, where $g(x)$ is some function we will solve for.

We compute:

• $f'(x) = e^{g(x)} \cdot g'(x)$,
• $f(f(x)) = e^{g(e^{g(x)})}$.

Step 4: Explore numerical or more sophisticated solutions

In general, such functional equations require a more advanced or numerical approach to determine if a simple closed-form solution exists.

General Insight:

At this stage, solving this equation explicitly may require more sophisticated techniques, or it may not have a simple elementary function as a solution. However, it’s known that certain forms of recursive or iterative functions might satisfy such relationships, and they often lead to more complex or transcendental solutions.

Questions for further exploration:

1. What would be the behavior of the solution $f(x)$ near specific points (e.g., at $x = 0$)?
2. Can you explore any known special functions that satisfy recursive functional equations like this one?
3. How might an iterative or numerical approach help solve this equation?
4. What properties should $f(x)$ have for $f'(x) = f(f(x))$ to hold for all $x$?
5. Are there any general methods in solving differential equations that are recursive or self-referential?

Tip: For solving complex functional equations, sometimes assuming a structure or pattern (like exponential or logarithmic forms) and testing specific properties can help narrow down the search.