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!