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The problem asks for the derivative of the polynomial function \( f(x) = x^5 - 3x^4 + 6 \) evaluated at \( x = 6 \).

Let's break it down step-by-step:

### 1. Find the derivative of the function:
The original function is:
\[
f(x) = x^5 - 3x^4 + 6
\]
Using the power rule for differentiation, \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \), we can differentiate each term.

- The derivative of \( x^5 \) is \( 5x^4 \).
- The derivative of \( -3x^4 \) is \( -12x^3 \).
- The derivative of the constant \( 6 \) is 0.

Thus, the derivative of the function is:
\[
f'(x) = 5x^4 - 12x^3
\]

### 2. Evaluate the derivative at \( x = 6 \):
Now, substitute \( x = 6 \) into the derivative function:
\[
f'(6) = 5(6)^4 - 12(6)^3
\]

Let's compute that.

I'll calculate it.The value of the derivative \( f'(6) \) is 3888.

Would you like more details, or do you have any other questions?

Here are five related questions to expand your understanding:
1. How would the result change if the polynomial had a quadratic term as well?
2. What is the second derivative of \( f(x) = x^5 - 3x^4 + 6 \)?
3. How does the power rule apply to non-integer exponents?
4. Can you find the derivative of a function using limits rather than rules?
5. How would you interpret the result of this derivative in terms of the slope of the tangent line?

**Tip**: Always remember to simplify polynomials before differentiating if possible; it can save time and avoid mistakes!

La derivada de la función polinomial f(x) = x^5 - 3x^4 + 6 en x = 6 es:

This webpage explains how to find the derivative of the polynomial function f(x) = x^5 - 3x^4 + 6, step by step, and evaluates the derivative at x = 6. The solution uses the power rule from calculus and discusses the fundamental theorems of derivatives. Suitable for students in Grades 10-12.

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