Sign of the Derivative

The sign of the derivative at a point tells us immediately whether the function is increasing or decreasing at that point. A positive derivative means the function rises; a negative derivative means it falls.

Condition	Meaning
f'(x_0) > 0	$f$ is increasing at $x_{0}$
f'(x_0) < 0	$f$ is decreasing at $x_{0}$
f'(x_0) = 0	tangent is horizontal at $x_{0}$

Interpreting the Sign

Negative velocity of an object moving on a straight line means that the object is moving in the negative direction. In other words, $v = \frac{d s}{d t} < 0$ means that $s$ is algebraically decreasing with time. Similarly, positive velocity means the object is moving in the positive direction and $s$ is increasing as $t$ increases.

The words "increase" and "decrease" are always used in the algebraic sense. For example, if a quantity changes from $- 8$ to $- 2$ , it algebraically increases, even though numerically it decreases.

Positive acceleration ( $a = \frac{d v}{d t} > 0$ ) means that the velocity $v$ is increasing as $t$ increases, and negative acceleration means that $v$ is decreasing as $t$ increases.

In general, if $y = f (x)$ , then:

if f'(x_0) > 0, an increase in the value of $x$ causes an increase in the value of $y$ .
if f'(x_0) < 0, an increase in the value of $x$ causes a decrease in the value of $y$ .

To be more precise, if f'(x_0) > 0, then $f$ is increasing at $x_{0}$ ; that is, for every $x$ sufficiently close to $x_{0}$ ,

f (x) < f (x_{0}) if x < x_{0}

and

f (x) > f (x_{0}) if x > x_{0} .

Similarly, if f'(x_0) < 0, then $f$ is decreasing at $x_{0}$ ; that is, for every $x$ sufficiently close to $x_{0}$ ,

f (x) > f (x_{0}) if x < x_{0}

and

f (x) < f (x_{0}) if x > x_{0} .

Theorem

Let $y = f (x)$ .

If f'(x_0) > 0, then $f$ is increasing at $x_{0}$ .
If f'(x_0) < 0, then $f$ is decreasing at $x_{0}$ .

Proof

Since f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}, if f'(x_0) > 0, then

Δ y / Δ x

must also be positive when

| Δ x |

is sufficiently small. This means that for small values of

Δ x

Δ y

and

Δ x

have the same sign: if

Δ x > 0

, then

Δ y > 0

as well, so

y_{0} + Δ y > y_{0}

. Graphically, when the slope of the tangent line is positive, the graph of

f

is rising at

x = x_{0}

. Similarly, if f'(x_0) < 0, then

Δ y

and

Δ x

must have opposite signs for sufficiently small

| Δ x |

. If

Δ x > 0

, then

y_{0} + Δ y < y_{0}

. Graphically, if f'(x_0) < 0, the graph of

f

is falling at

x = x_{0}

Frequently Asked Questions

Can a function be increasing at a point where the derivative is zero?

No. If f'(x_0) = 0, the theorem says neither "increasing" nor "decreasing" at

x_{0}

in the strict sense of the definition. The function has a horizontal tangent there and may have a local extremum, or it may continue in the same direction (as with

f (x) = x^{3}

x = 0

Is the converse of the theorem true?

Not in general. If

f

is increasing at

x_{0}

, it does not necessarily follow that f'(x_0) > 0. For example,

f (x) = x^{3}

is increasing at

x = 0

but f'(0) = 0.

Interpreting the Sign

Theorem

Frequently Asked Questions

Quick Links

Social Media