What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?

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Question: What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?

Options:

Correct Answer: 4x^3 - 12x^2 + 12x - 4

Solution:

The derivative is f\'(x) = 4x^3 - 12x^2 + 12x - 4.

What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?

What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?

Step 1: Identify the function f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
Step 2: Recall the power rule for derivatives: If f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Apply the power rule to each term in the function.
Step 4: For the first term x^4, the derivative is 4*x^(4-1) = 4x^3.
Step 5: For the second term -4x^3, the derivative is -4*3*x^(3-1) = -12x^2.
Step 6: For the third term 6x^2, the derivative is 6*2*x^(2-1) = 12x.
Step 7: For the fourth term -4x, the derivative is -4*1*x^(1-1) = -4.
Step 8: The last term is a constant (1), and the derivative of a constant is 0.
Step 9: Combine all the derivatives from Steps 4 to 8: f'(x) = 4x^3 - 12x^2 + 12x - 4.

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