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We are tasked with finding the derivative \( f'(x) \) for the function:

\[
f(x) = 6x - 4x^9 + 16e^x
\]

To differentiate this, we apply the following rules:
1. **Power Rule** for \( x^n \): The derivative of \( x^n \) is \( nx^{n-1} \).
2. **Constant Rule**: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
3. **Exponential Rule**: The derivative of \( e^x \) is simply \( e^x \).

### Step-by-step differentiation:

1. Derivative of \( 6x \): 
   \[
   \frac{d}{dx}(6x) = 6
   \]
   
2. Derivative of \( -4x^9 \):
   \[
   \frac{d}{dx}(-4x^9) = -4 \cdot 9x^{8} = -36x^8
   \]
   
3. Derivative of \( 16e^x \):
   \[
   \frac{d}{dx}(16e^x) = 16e^x
   \]

### Putting it all together:

\[
f'(x) = 6 - 36x^8 + 16e^x
\]

This is the derivative of the given function.

---

Would you like a more detailed explanation of any step? Here are 5 related questions to explore further:

1. How does the Power Rule work for negative exponents?
2. What are the derivatives of common exponential functions besides \( e^x \)?
3. Can the Chain Rule be applied if the function involves more complex terms?
4. What happens if we integrate the function \( f(x) \) instead?
5. How does the derivative change if the constants in the original function are modified?

**Tip:** Always remember to simplify your derivatives after applying the rules to avoid small errors in larger problems!

Find f'(x) for the function f(x) = 6x - 4x^9 + 16e^x.

Learn how to find the derivative of f(x) = 6x - 4x^9 + 16e^x using the Power Rule and Exponential Rule. This problem covers key differentiation concepts, including polynomial and exponential functions. Suitable for advanced high school calculus students.

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