What is the derivative of f(x) = x^3?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

What is the derivative of f(x) = x^3?

Explanation:
Differentiating a power of x uses the power rule: d/dx x^n = n x^{n-1}. For x^3, that gives 3x^2. So the rate of change (the slope of the tangent) to y = x^3 at any x is 3x^2, which grows with the square of x. The other expressions don’t fit this rule for n = 3: 2x comes from differentiating x^2, x^2 is the original function, and 3x would correspond to a different power, not x^3.

Differentiating a power of x uses the power rule: d/dx x^n = n x^{n-1}. For x^3, that gives 3x^2. So the rate of change (the slope of the tangent) to y = x^3 at any x is 3x^2, which grows with the square of x. The other expressions don’t fit this rule for n = 3: 2x comes from differentiating x^2, x^2 is the original function, and 3x would correspond to a different power, not x^3.

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