Reading the Derivative from a Graph

One rule reads the sign of a derivative straight off a graph: flat means zero, rising means positive, falling means negative. Five very different functions - a constant, a line, a parabola, a cube and eˣ - all obey it, and the same pictures show why a curve can sit above the x-axis and still have a negative derivative.

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جارٍ تحميل الفيديو…

You do not need the equation. Look at the curve: flat means the derivative is zero, rising means it is positive, falling means it is negative. That is the whole rule — and five functions that could hardly look more different all obey it.

Concept The whole rule
Curve is flat  ⟹   (no change)
Curve is rising  ⟹  f'(x) > 0
Curve is falling  ⟹  f'(x) < 0

The derivative only asks which way the curve is heading. How high the curve sits — above the axis or below it — has nothing to do with the sign of .

Concept Five functions, one rule
— a horizontal line, no change anywhere. .
— climbs by the same amount at every step, a constant slope of 1. .
− 0 + x²
— the left branch is falling, so is negative there; the vertex is flat, so ; the right branch is rising, so is positive. A value that runs from negative, through zero, to positive as increases is exactly .
— rising on the left even while it sits below the axis, flat for an instant at the origin, then rising again. It never falls, which is why its derivative is never negative.
— almost flat far to the left, so is close to 0; steeper and steeper moving right, so grows without bound. At every point the steepness equals the height, and that is why — the one function that is its own derivative.
Note Height is not slope

Two of these examples make the same point. On the left branch of the curve is well above the -axis, yet it is falling — so the derivative there is negative. On the left part of the curve is below the axis, yet it is rising — so the derivative there is positive. Where the curve sits tells you the value of the function; which way it leans tells you the value of the derivative. They are different questions.

Summary
  1. Flat ⟹ ; rising ⟹ f' > 0; falling ⟹ f' < 0.
  2. gives ; gives .
  3. falls, flattens, then rises — matching running negative, zero, positive.
  4. rises, flattens, rises — so is never negative.
  5. is its own derivative.
  6. A curve's height above or below the axis says nothing about the sign of its derivative.
283. Relations and Functions
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