Just use simple math and derivative rules

Certainly! Let's solve this using basic derivative rules.

Given the equation:

\[
f(x + e) = 5x^2 \cdot e + 2x \cdot e^2 + f(x)
\]

We want to find the first derivative \( f'(x) \) when \( x = 2 \).

### Step 1: Differentiate both sides with respect to \( e \)

We can differentiate the given equation with respect to \( e \) and then set \( e = 0 \) to find the derivative of \( f(x) \).

Differentiating both sides with respect to \( e \):

\[
\frac{d}{de}f(x + e) = \frac{d}{de}\left(5x^2 \cdot e + 2x \cdot e^2 + f(x)\right)
\]

On the left side, using the chain rule:

\[
f'(x + e) \cdot \frac{d(x + e)}{de} = f'(x + e) \cdot 1
\]

On the right side:

\[
\frac{d}{de}(5x^2 \cdot e) + \frac{d}{de}(2x \cdot e^2) + \frac{d}{de}(f(x))
\]

Since \( f(x) \) is independent of \( e \), its derivative with respect to \( e \) is zero:

\[
f'(x + e) = 5x^2 + 4x \cdot e
\]

### Step 2: Set \( e = 0 \)

To find \( f'(x) \), we set \( e = 0 \):

\[
f'(x) = 5x^2 + 4x \cdot 0 = 5x^2
\]

### Step 3: Substitute \( x = 2 \)

Now, substitute \( x = 2 \):

\[
f'(2) = 5(2)^2 = 5 \cdot 4 = 20
\]

### Answer:
The first derivative of \( f(x) \) at \( x = 2 \) is \( f'(2) = 20 \).

Would you like to go over any specific part of the solution?

Here are some related questions:
1. What would be \( f'(x) \) if the equation were \( f(x+e) = 3x^2 \cdot e + f(x) \)?
2. How do you apply the chain rule to differentiate composite functions?
3. What happens if \( e \) is not set to zero after differentiating?
4. How do you verify if a function is differentiable?
5. How does the value of \( f'(x) \) change as \( x \) increases?
6. How would you find \( f''(x) \) using the same method?
7. Can you generalize this method to higher dimensions?
8. How does this approach relate to the concept of a limit?

**Tip:** When differentiating equations involving small changes, always remember to evaluate at the point of interest to get the specific derivative value.

Explore how to solve a simple math problem using derivative rules and the chain rule. Learn step-by-step how to differentiate a function involving small changes. Suitable for high school students studying calculus.

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