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The problem states:

Given \( f(x) = x^5 - 4x^2 + 8 \), find the value of \( f'(1) \).

### Solution:

1. **Differentiate \( f(x) \) with respect to \( x \):**

   The derivative \( f'(x) \) of \( f(x) = x^5 - 4x^2 + 8 \) is found by applying the power rule:

   \[
   f'(x) = \frac{d}{dx}(x^5) - \frac{d}{dx}(4x^2) + \frac{d}{dx}(8)
   \]

   - The derivative of \( x^5 \) is \( 5x^4 \).
   - The derivative of \( -4x^2 \) is \( -8x \).
   - The derivative of the constant \( 8 \) is \( 0 \).

   So,

   \[
   f'(x) = 5x^4 - 8x
   \]

2. **Substitute \( x = 1 \) into \( f'(x) \):**

   \[
   f'(1) = 5(1)^4 - 8(1)
   \]

   Simplifying,

   \[
   f'(1) = 5 - 8 = -3
   \]

### Answer:
The value of \( f'(1) \) is \(-3\).

Would you like a further breakdown of each step, or do you have any other questions?

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Here are five related questions for further exploration:

1. What is the second derivative \( f''(x) \) of \( f(x) = x^5 - 4x^2 + 8 \)?
2. How would you find \( f'(x) \) for a more complex polynomial, like \( x^6 - 3x^4 + 5x^3 \)?
3. How does the power rule help simplify derivatives of polynomial functions?
4. What is the significance of \( f'(x) = 0 \) in understanding the behavior of a function?
5. How would the process change if \( f(x) \) included a trigonometric term like \( \sin(x) \)?

**Tip:** Always start by differentiating each term individually and carefully apply the power rule to each

Diberikan f(x) = x^5 - 4x^2 + 8. Berapakah nilai f'(1)?

Learn how to find the derivative f'(1) for the function f(x) = x^5 - 4x^2 + 8. This calculus problem involves differentiation using the power rule and evaluates the result at a specific point. Suitable for high school students in Grades 11-12.

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