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 ⟹

f'(x) = 0

(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 $f'(x)$ .

Concept Five functions, one rule

$y = 5$ — a horizontal line, no change anywhere.

f'(x) = 0

$y = x$ — climbs by the same amount at every step, a constant slope of 1.

f'(x) = 1

$y = x^{2}$ — the left branch is falling, so

f'

is negative there; the vertex is flat, so

f' = 0

; the right branch is rising, so

f'

is positive. A value that runs from negative, through zero, to positive as

x

increases is exactly

f'(x) = 2x

$y = x^{3}$ — 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

f'(x) = 3x^{2}

is never negative.

$y = e^{x}$ — almost flat far to the left, so

f'

is close to 0; steeper and steeper moving right, so

f'

grows without bound. At every point the steepness equals the height, and that is why

f'(x) = e^{x}

— 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 $y = x^{2}$ the curve is well above the $x$ -axis, yet it is falling — so the derivative there is negative. On the left part of $y = x^{3}$ 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

Flat ⟹ $f' = 0$ ; rising ⟹ $f' > 0$ ; falling ⟹ $f' < 0$ .
$y = 5$ gives $f'(x) = 0$ ; $y = x$ gives $f'(x) = 1$ .
$y = x^{2}$ falls, flattens, then rises — matching $f'(x) = 2x$ running negative, zero, positive.
$y = x^{3}$ rises, flattens, rises — so $f'(x) = 3x^{2}$ is never negative.
$y = e^{x}$ is its own derivative.
A curve's height above or below the axis says nothing about the sign of its derivative.

Reading the Derivative from a Graph