The Power Rule for Polynomials
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Flip to reveal answersState the power rule for differentiation.
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All 8 Flashcards — The Power Rule for Polynomials
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Question
State the power rule for differentiation.
Answer
d/dx[axⁿ] = naxⁿ⁻¹. Multiply the coefficient by the power, then reduce the power by one.
Question
Differentiate f(x) = 5x⁴.
Answer
f′(x) = 20x³. (Multiply 5 by 4 = 20, reduce power from 4 to 3.)
Question
What is d/dx[8]?
Answer
0. The derivative of any constant is zero.
Question
What is d/dx[−7x]?
Answer
−7. The derivative of ax is a. Here a = −7.
Question
Find f′(x) for f(x) = 3x³ − 2x² + x − 9.
Answer
f′(x) = 9x² − 4x + 1. Apply the power rule to each term. The constant −9 disappears. The linear x term gives 1.
Question
Before differentiating y = x(4x − 1), what must you do first?
Answer
Expand: y = 4x² − x. Then differentiate: dy/dx = 8x − 1. You cannot apply the power rule inside a product without expanding.
Question
Find the gradient of y = 2x³ − x at x = 2.
Answer
dy/dx = 6x² − 1. At x = 2: 6(4) − 1 = 23.
💡 Hint
Differentiate first, then substitute.
Question
For f(x) = x², you get f(3) = 9 and f′(3) = 6. What does each number represent?
Answer
f(3) = 9 is the y-value of the curve at x = 3. f′(3) = 6 is the gradient of the curve at x = 3. Different quantities with different meanings.
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