If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(2).

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Question: If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f\'(2).

Options:

Correct Answer: 0

Solution:

f\'(x) = 4x^3 - 12x^2 + 12x - 4; f\'(2) = 0.

If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(2).

If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(2).

Step 1: Identify the function f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
Step 2: Find the derivative of the function, f'(x).
Step 3: Use the power rule to differentiate each term: The derivative of x^n is n*x^(n-1).
Step 4: Differentiate each term: 4x^3 (from x^4), -12x^2 (from -4x^3), 12x (from 6x^2), and -4 (from -4x).
Step 5: Combine the derivatives to get f'(x) = 4x^3 - 12x^2 + 12x - 4.
Step 6: Substitute x = 2 into the derivative: f'(2) = 4(2)^3 - 12(2)^2 + 12(2) - 4.
Step 7: Calculate each term: 4(8) = 32, -12(4) = -48, 12(2) = 24, and -4 remains -4.
Step 8: Combine the results: 32 - 48 + 24 - 4 = 0.
Step 9: Conclude that f'(2) = 0.

Differentiation – The process of finding the derivative of a function.
Polynomial Functions – Understanding the behavior and properties of polynomial functions.
Evaluation of Derivatives – Calculating the value of the derivative at a specific point.