Introduction to derivatives
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What does the derivative f′(x) tell you?
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All Flashcards in Topic 5.3
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5.3.18 cards
What does the derivative f′(x) tell you?
f′(x) is the gradient function. It gives the gradient of the curve y = f(x) at any x-value. Substitute a number into f′(x) to get the gradient at that point.
Think: steepness, not height.
What does the notation dy/dx mean?
dy/dx is "the derivative of y with respect to x". It is exactly the same thing as f′(x). Both notations appear in IB papers.
What is the sign of f′(x) when the curve is rising?
f′(x) > 0 when the curve is increasing (rising left to right). f′(x) < 0 when decreasing. f′(x) = 0 at a local maximum or minimum.
A curve has a local maximum at x = 3. What is f′(3)?
f′(3) = 0. At any local maximum (or minimum), the tangent is horizontal, so the gradient is zero.
Flat tangent = zero gradient.
Why does a straight line NOT need differentiation to find its gradient?
A straight line has the same gradient everywhere. For y = mx + c, the gradient is always m. Only curves have a different gradient at each point.
V(t) is the volume (litres) in a tank. What does V′(t) = −5 mean?
The volume is decreasing at a rate of 5 litres per unit time. The negative sign means the function is falling. Always include units in your interpretation.
Rate of change — always state units.
What is the difference between f(a) and f′(a)?
f(a) is the y-value (height) of the curve at x = a.\nf′(a) is the gradient (steepness) of the curve at x = a.\nThey are completely different quantities.
A curve is high up on the graph (large y-value) at x = 5, but f′(5) = 0. Is that possible?
Yes. f(x) and f′(x) are independent. A curve can be at any height while being momentarily flat — for example, at the top of a hill.
5.3.28 cards
State the power rule for differentiation.
d/dx[axⁿ] = naxⁿ⁻¹. Multiply the coefficient by the power, then reduce the power by one.
Differentiate f(x) = 5x⁴.
f′(x) = 20x³. (Multiply 5 by 4 = 20, reduce power from 4 to 3.)
What is d/dx[8]?
0. The derivative of any constant is zero.
What is d/dx[−7x]?
−7. The derivative of ax is a. Here a = −7.
Find f′(x) for f(x) = 3x³ − 2x² + x − 9.
f′(x) = 9x² − 4x + 1. Apply the power rule to each term. The constant −9 disappears. The linear x term gives 1.
Before differentiating y = x(4x − 1), what must you do first?
Expand: y = 4x² − x. Then differentiate: dy/dx = 8x − 1. You cannot apply the power rule inside a product without expanding.
Find the gradient of y = 2x³ − x at x = 2.
dy/dx = 6x² − 1. At x = 2: 6(4) − 1 = 23.
Differentiate first, then substitute.
For f(x) = x², you get f(3) = 9 and f′(3) = 6. What does each number represent?
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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