Math Problem Statement

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

Solution

The problem states:

Given f(x)=x54x2+8f(x) = x^5 - 4x^2 + 8, find the value of f(1)f'(1).

Solution:

  1. Differentiate f(x)f(x) with respect to xx:

    The derivative f(x)f'(x) of f(x)=x54x2+8f(x) = x^5 - 4x^2 + 8 is found by applying the power rule:

    f(x)=ddx(x5)ddx(4x2)+ddx(8)f'(x) = \frac{d}{dx}(x^5) - \frac{d}{dx}(4x^2) + \frac{d}{dx}(8)

    • The derivative of x5x^5 is 5x45x^4.
    • The derivative of 4x2-4x^2 is 8x-8x.
    • The derivative of the constant 88 is 00.

    So,

    f(x)=5x48xf'(x) = 5x^4 - 8x

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

    f(1)=5(1)48(1)f'(1) = 5(1)^4 - 8(1)

    Simplifying,

    f(1)=58=3f'(1) = 5 - 8 = -3

Answer:

The value of f(1)f'(1) is 3-3.

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


Here are five related questions for further exploration:

  1. What is the second derivative f(x)f''(x) of f(x)=x54x2+8f(x) = x^5 - 4x^2 + 8?
  2. How would you find f(x)f'(x) for a more complex polynomial, like x63x4+5x3x^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)=0f'(x) = 0 in understanding the behavior of a function?
  5. How would the process change if f(x)f(x) included a trigonometric term like sin(x)\sin(x)?

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

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Power Rule

Formulas

f'(x) = nx^(n-1) for each term in polynomial differentiation

Theorems

Power Rule for Derivatives

Suitable Grade Level

Grades 11-12