What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)

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Question: What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)

Options:

Correct Answer: 4x^3 - 12x

Exam Year: 2022

Solution:

Using the power rule, f\'(x) = 4x^3 - 12x.

What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)

What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)

Step 1: Identify the function f(x) = x^4 - 6x^2 + 9.
Step 2: Recognize that we need to find the derivative, which tells us the rate of change of the function.
Step 3: Use the power rule for derivatives, which states that if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 4: Apply the power rule to each term in the function:
- For the first term x^4, the derivative is 4*x^(4-1) = 4x^3.
- For the second term -6x^2, the derivative is -6*2*x^(2-1) = -12x.
- For the constant term 9, the derivative is 0 because the derivative of a constant is always 0.
Step 5: Combine the derivatives of all terms: f'(x) = 4x^3 - 12x + 0.
Step 6: Simplify the expression: f'(x) = 4x^3 - 12x.

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