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To find the Maclaurin series for the function \( f(x) = 9x e^{-4x} \), we start by recalling that a Maclaurin series expands a function around \( x = 0 \). Let's break it down step-by-step:

1. **Separate the Function**: We have \( f(x) = 9x \cdot e^{-4x} \). We can treat \( 9x \) and \( e^{-4x} \) as two parts and expand \( e^{-4x} \) into its own Maclaurin series.

2. **Maclaurin Series for \( e^{-4x} \)**: The exponential function \( e^u \) has a Maclaurin series:
   \[
   e^u = \sum_{n=0}^{\infty} \frac{u^n}{n!}
   \]
   Here, we substitute \( u = -4x \), giving:
   \[
   e^{-4x} = \sum_{n=0}^{\infty} \frac{(-4x)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-4)^n x^n}{n!}
   \]

3. **Multiply by \( 9x \)**: Now, we multiply this series by \( 9x \):
   \[
   f(x) = 9x \cdot \sum_{n=0}^{\infty} \frac{(-4)^n x^n}{n!} = \sum_{n=0}^{\infty} \frac{9 \cdot (-4)^n x^{n+1}}{n!}
   \]

4. **Rewrite in Summation Form**: To put this in a standard summation form, let’s rewrite it as:
   \[
   f(x) = \sum_{n=0}^{\infty} \frac{9 \cdot (-4)^n}{n!} x^{n+1}
   \]

This is the Maclaurin series for \( f(x) = 9x e^{-4x} \). If there is an error in your answer, it could be due to the structure of the summation or terms not aligning with your solution's requirements.

Would you like further steps or clarification on this process?

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### Related Questions:
1. How do you find the Maclaurin series of more complex functions involving \( e^x \) and trigonometric terms?
2. What is the general form of the Maclaurin series for a function \( f(x) = g(x) \cdot h(x) \)?
3. How does multiplying by a polynomial affect the convergence of a Maclaurin series?
4. What are the applications of Maclaurin series in physics or engineering?
5. How can you verify the convergence of a Maclaurin series for different functions?

### Tip:
Always remember to separate each part of a function when calculating Maclaurin series for products, and use known series expansions to simplify calculations.

Find the Maclaurin series for f(x) = 9x e^{-4x}.

Calculate the Maclaurin series for the function f(x) = 9x * e^(-4x). This solution covers the power series expansion for e^(-4x), multiplication of series, and the steps to form the final series representation. Ideal for college-level calculus students.

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