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What is the derivative of f(x) = x^3 - 6x^2 + 9x?

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Question: What is the derivative of f(x) = x^3 - 6x^2 + 9x?

Options:

  1. 3x^2 - 12x + 9
  2. 3x^2 - 6x + 9
  3. 6x - 12
  4. 3x^2 - 9

Correct Answer: 3x^2 - 12x + 9

Solution: 

The derivative f\'(x) = 3x^2 - 12x + 9.

What is the derivative of f(x) = x^3 - 6x^2 + 9x?

Practice Questions

Q1
What is the derivative of f(x) = x^3 - 6x^2 + 9x?
  1. 3x^2 - 12x + 9
  2. 3x^2 - 6x + 9
  3. 6x - 12
  4. 3x^2 - 9

Questions & Step-by-Step Solutions

What is the derivative of f(x) = x^3 - 6x^2 + 9x?
  • Step 1: Identify the function f(x) = x^3 - 6x^2 + 9x.
  • Step 2: Use the power rule for derivatives, which states that 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^3, the derivative is 3*x^(3-1) = 3x^2.
  • Step 5: For the second term -6x^2, the derivative is -6*2*x^(2-1) = -12x.
  • Step 6: For the third term 9x, the derivative is 9*1*x^(1-1) = 9.
  • Step 7: Combine all the derivatives from each term: f'(x) = 3x^2 - 12x + 9.
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