If f(x+e)= 5x^2*e + 2x e^2 + f(x)
Find the first derivative of f(x) when x = 2

Given the functional equation:

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

we want to find the first derivative of \( f(x) \) when \( x = 2 \).

### Step 1: Approximate the function for small \( e \)
For small \( e \), we can use the following approximation:

\[
f(x + e) \approx f(x) + f'(x) \cdot e + \frac{1}{2}f''(x) \cdot e^2
\]

### Step 2: Compare the given equation with the Taylor expansion
Now, compare the given equation with the Taylor expansion:

\[
f(x) + f'(x) \cdot e + \frac{1}{2}f''(x) \cdot e^2 \approx 5x^2 \cdot e + 2x \cdot e^2 + f(x)
\]

### Step 3: Match the coefficients of \( e \) and \( e^2 \)
- Coefficient of \( e \):
  \[
  f'(x) = 5x^2
  \]

- Coefficient of \( e^2 \):
  \[
  \frac{1}{2}f''(x) = 2x
  \]
  \[
  f''(x) = 4x
  \]

### Step 4: Find the first derivative at \( x = 2 \)
Substitute \( x = 2 \) into the first derivative:

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

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

Would you like further details or explanations?

Here are some related questions to consider:
1. How would you find the second derivative of \( f(x) \) at \( x = 2 \)?
2. What is the significance of matching coefficients in a Taylor series?
3. How does the Taylor series expansion help in approximating functions?
4. What if the given equation included higher powers of \( e \)?
5. How would the result change if the coefficient of \( e^2 \) was different?
6. How can you verify the correctness of the obtained derivative?
7. Can you apply a similar method for finding higher-order derivatives?
8. What are the limitations of this method for approximating derivatives?

**Tip:** When dealing with functional equations involving small increments, the Taylor series expansion is a powerful tool for finding derivatives.

Explore the solution for finding the first derivative of f(x) given a functional equation using Taylor series approximation. Learn step-by-step how to match coefficients and compute derivatives effectively. Suitable for advanced high school and undergraduate students.

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