Math Problem Statement

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

Solution

Given the function f(x)=x75x5+4x33x8f(x) = x^7 - 5x^5 + 4x^3 - 3x - 8, the results are as follows:

  1. f(x)=7x625x4+12x23f'(x) = 7x^6 - 25x^4 + 12x^2 - 3
  2. f(4)=22,461f'(4) = 22,461
  3. f(x)=42x5100x3+24xf''(x) = 42x^5 - 100x^3 + 24x
  4. f(4)=36,704f''(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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Polynomial Functions

Formulas

Power Rule: d/dx [x^n] = n*x^(n-1)
Derivative of a constant is zero

Theorems

Basic Differentiation Rules
Higher-Order Derivatives

Suitable Grade Level

Grades 11-12 (Advanced Calculus)