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To solve this, we need to find the 51st derivative of the given function:

\[
f(x) = e^{2x} + x^{20}
\]

### Step-by-Step Solution:

#### 1. Derivatives of \(e^{2x}\):
The function \(e^{2x}\) has a simple pattern when differentiated. Each derivative brings down a factor of 2, so:
- First derivative: \(f'(x) = 2e^{2x}\)
- Second derivative: \(f''(x) = 4e^{2x}\)
- Third derivative: \(f^{(3)}(x) = 8e^{2x}\)
- And in general, the \(n\)-th derivative of \(e^{2x}\) is:
  \[
  f^{(n)}(x) = 2^n e^{2x}
  \]
So the 51st derivative of \(e^{2x}\) is:
\[
f^{(51)}(x) = 2^{51} e^{2x}
\]

#### 2. Derivatives of \(x^{20}\):
The term \(x^{20}\) will vanish after taking 21 derivatives:
- First derivative: \(20x^{19}\)
- Second derivative: \(20 \cdot 19x^{18}\)
- Continuing this process, after 20 derivatives, the result will be:
  \[
  f^{(20)}(x) = 20!
  \]
- The 21st derivative will be 0, and all higher derivatives of \(x^{20}\) (including the 51st) will also be 0:
  \[
  f^{(n)}(x) = 0 \quad \text{for} \quad n \geq 21
  \]

### Final Answer:
The 51st derivative of \(f(x) = e^{2x} + x^{20}\) is the 51st derivative of \(e^{2x}\) plus the 51st derivative of \(x^{20}\):
\[
f^{(51)}(x) = 2^{51} e^{2x} + 0 = 2^{51} e^{2x}
\]

Would you like more details on any step or have any further questions?

Here are 5 related questions to consider:
1. How do the rules of differentiation apply to more complex exponential functions?
2. Why do polynomials eventually have zero derivatives after enough differentiation?
3. How does the chain rule affect the differentiation of functions like \(e^{2x}\)?
4. What would happen if the exponent on \(x\) was higher than 20 in the given polynomial?
5. Can we use a general formula to find the \(n\)-th derivative of more complex functions?

**Tip**: When differentiating terms like \(x^n\), remember that after taking \(n+1\) derivatives, the result will always be zero!

What is the 51st derivative of f(x) = e^2x + x^20?

Learn how to find the 51st derivative of the function f(x) = e^2x + x^20. This calculus problem covers key differentiation concepts such as exponential functions and polynomial functions, utilizing the power rule and chain rule for higher derivatives. Suitable for advanced high school or college-level calculus students.

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