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Given the function \( f(x) = x^7 - 5x^5 + 4x^3 - 3x - 8 \), the results are as follows:

1. \( f'(x) = 7x^6 - 25x^4 + 12x^2 - 3 \)
2. \( f'(4) = 22,461 \)
3. \( f''(x) = 42x^5 - 100x^3 + 24x \)
4. \( f''(4) = 36,704 \)

Would you like more details on any of these derivatives or calculations?

Here are 5 related questions to consider:
1. How is the first derivative related to the slope of the function at a given point?
2. What does the second derivative tell you about the concavity of the function?
3. How do you compute higher-order derivatives for complex functions?
4. Why are critical points important in the context of derivatives?
5. How would you apply the chain rule if the function involved composite expressions?

**Tip**: Always double-check your derivatives step by step, especially when differentiating terms with multiple powers.

Let f(x) = x^7 - 5x^5 + 4x^3 - 3x - 8. Find f'(x), f'(4), f''(x), and f''(4).

Learn how to find the first and second derivatives of the polynomial function f(x) = x^7 - 5x^5 + 4x^3 - 3x - 8. Includes step-by-step solutions for f'(x), f'(4), f''(x), and f''(4), with detailed calculations. Suitable for advanced calculus students.

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