What is the derivative function?

The derivative function f'(x) is defined by the limit of the difference quotient evaluated at variable x rather than a fixed point. Wherever this limit exists and is finite, f'(x) gives the slope of f at x. The domain of f' is all points where f is differentiable.

How do you find the derivative function from the definition?

Apply the limit f'(x) = lim(h→0) [f(x+h) - f(x)]/h with x as a free variable. For f(x) = x², expanding gives (2xh + h²)/h = 2x + h, and the limit is 2x. The result is a function that outputs the slope at any input x.

How do you graph f' from the graph of f?

Where f is increasing, f' is positive (above x-axis). Where f is decreasing, f' is negative. Steep sections give large |f'|; flat sections give f' near zero. Local extrema of f correspond to zeros of f'. Concavity of f determines whether f' is increasing or decreasing.

How do you reconstruct f from f'?

Where f' is positive, f rises; where f' negative, f falls. Sign changes of f' indicate extrema of f. The magnitude of f' controls steepness. Where f' is increasing, f is concave up. The vertical position of f is undetermined—any constant can be added.

How is the derivative a rate of change function?

When f models a varying quantity, f'(x) models how fast it changes at each x. For position s(t), the derivative s'(t) is velocity as a function of time. For cost C(x), the derivative C'(x) is marginal cost as a function of production level.

Must the derivative function be continuous?

No. A function can be differentiable everywhere yet have a discontinuous derivative. However, Darboux's theorem says f' always has the intermediate value property—it cannot have jump discontinuities, only oscillatory ones. Functions with continuous derivatives form class C¹.

Derivatives as Functions

Q: What is the relationship between f and f'?

Zeros of f' correspond to horizontal tangents on f. Sign changes of f' indicate direction changes of f. Extrema of f' correspond to inflection points of f. The derivative encodes the complete shape of f up to vertical translation.

The Derivative Function Defined

Computing the Derivative Function from the Definition

Graphing f' from f

Graphing f from f'

The Relationship Between f and f'

The Derivative as Rate of Change

Continuity of the Derivative

Summary: How f and f' Correspond

From a Single Slope to an Entire Function

The derivative at a point

f'(a)

is a number—one slope at one location. But the definition works at every point where the limit exists, and letting the input vary produces a new function

f'(x)

that maps each

x

to the slope of

f

at that

x

.

This shift in perspective is fundamental. The derivative function

f'

has its own domain, its own graph, its own behavior. It can be continuous or discontinuous, positive or negative, increasing or decreasing. It can itself be differentiated, producing higher-order derivatives.

Understanding

f'

as a function—not just a computation applied at isolated points—connects the local information encoded in slopes to global information about the shape of

f

Key Terms

Derivative— the limit of the difference quotient, producing a slope at each point

Instantaneous Rate of Change— the derivative interpreted as a rate

Average Rate of Change— secant slope over an interval

Differentiability— determines the domain of the derivative function

Monotonic Function— sign of

f'

determines where

f

rises or falls

See All Calculus Definitions →

Computing the Derivative Function from the Definition

Applying the limit definition with

x

as a free variable produces an expression in

x

—the derivative function in closed form.

For

f(x) = x^2

f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x

The result

f'(x) = 2x

is a function: input

x

, output the slope of the parabola at that

x

. At

x = 3

, the slope is

6

. At

x = -1

, the slope is

-2

.

Once differentiation rules are established, this limit computation becomes unnecessary for standard functions. But the limit definition remains the foundation—every rule is proven from it, and nonstandard functions may require returning to it directly.

Graphing f' from f

The graph of

f'

can be read directly from the graph of

f

without computing any formula.

Where

f

is increasing,

f'

is positive—the graph of

f'

lies above the

x

-axis. Where

f

is decreasing,

f'

is negative—the graph of

f'

lies below the

x

-axis. Where

f

has a local extremum,

f'

crosses zero (or is undefined).

Steep sections of

f

correspond to large values of

|f'|

. Flat sections of

f

correspond to

f'

near zero. A straight segment of

f

with constant slope corresponds to

f'

being constant—a horizontal segment on the graph of

f'

.

Where

f

is concave up, the slope is increasing, so

f'

is an increasing function on that interval. Where

f

is concave down,

f'

is decreasing. Inflection points of

f

—where concavity changes—correspond to local extrema of

f'

Graphing f from f'

The reverse problem—reconstructing

f

from

f'

—is possible up to a vertical shift.

Where

f'

is positive,

f

is rising. Where

f'

is negative,

f

is falling. Where

f'

crosses zero from positive to negative,

f

has a local maximum. Where

f'

crosses zero from negative to positive,

f

has a local minimum.

The magnitude of

f'

controls steepness: large positive values of

f'

mean

f

is climbing steeply; values of

f'

near zero mean

f

is nearly flat. Where

f'

is increasing,

f

is concave up. Where

f'

is decreasing,

f

is concave down.

The one piece of information

f'

does not determine is the vertical position. The derivative of

f(x) = x^2

and the derivative of

g(x) = x^2 + 5

are identical: both equal

2x

. The graph of

f

can be shifted vertically by any constant without changing

f'

. This ambiguity is resolved only when a specific function value is known—an initial condition.

The Relationship Between f and f'

The derivative function

f'

encodes the shape of

f

completely, up to vertical translation. The correspondence runs in both directions:

Zeros of

f'

correspond to horizontal tangents on

f

—local extrema or plateaus. Sign changes of

f'

correspond to direction changes of

f

. Extrema of

f'

correspond to inflection points of

f

, where the bending direction reverses.

If

f'

is itself a smooth, well-behaved function, then

f

is smooth as well. If

f'

has jumps or singularities, those indicate abrupt changes in the slope of

f

—corners, cusps, or worse.

This relationship extends to higher-order derivatives. The function

f''

relates to

f'

in the same way that

f'

relates to

f

: the second derivative is the derivative of the derivative, and the correspondence between zeros, sign changes, and extrema repeats at each level.

The Derivative as Rate of Change

When

f

models a quantity that varies with input

x

, the derivative function

f'(x)

models how fast that quantity changes at each value of

x

.

In kinematics, if

s(t)

is position as a function of time, then

s'(t)

is velocity—a function that gives the speed and direction of motion at every instant. The velocity function can be graphed, analyzed for zeros (stops), sign changes (reversals), and extrema (maximum speed). Differentiating again yields

s''(t)

, acceleration, which is itself a function of time.

In economics, if

C(x)

is the cost of producing

x

units, then

C'(x)

is the marginal cost—the rate at which cost changes per additional unit. The marginal cost function reveals whether production costs are accelerating or stabilizing.

In each context, the derivative function provides a running commentary on how the original quantity evolves—not at a single frozen moment, but across the entire domain.

Context	Quantity f	Derivative f'	What f' reveals
General	quantity as a function of x	instantaneous rate of change at each x	where the quantity accelerates, plateaus, or reverses
Kinematics	position s(t)	velocity s'(t)	stops (zeros), reversals (sign changes), maximum speed (extrema)
Economics	total cost C(x)	marginal cost C'(x)	whether production costs are accelerating or stabilizing

Continuity of the Derivative

A natural question: if

f

is differentiable everywhere, must

f'

be continuous?

The answer is no. Consider

f(x) = x^2 \sin(1/x)

for

x \neq 0

and

f(0) = 0

. This function is differentiable at every real number, including

x = 0

, yet

f'

oscillates near the origin and is discontinuous there. Differentiability of

f

does not guarantee continuity of

f'

.

However, derivatives satisfy a strong structural constraint. Darboux's theorem states that

f'

always has the intermediate value property: if

f'

takes values

A

and

B

on an interval, it takes every value between

A

and

B

somewhere on that interval. This means

f'

cannot have jump discontinuities. The only discontinuities possible for a derivative are oscillatory ones—the kind seen in the example above.

Functions whose derivatives are continuous everywhere form the class

C^1

. Most functions encountered in standard calculus belong to

C^1

or higher smoothness classes, but the distinction matters in theoretical work and in the study of differentiability.

Class	Definition	Behavior of f'	Example
Continuous (C⁰)	f continuous on its domain	may not exist at every point	\|x\| — continuous, not differentiable at 0
Differentiable	f' exists at every point of the domain	exists, but may itself be discontinuous	x² sin(1/x) extended by f(0)=0
Continuously differentiable (C¹)	f differentiable and f' continuous	continuous on the whole domain	polynomials, sin x, eˣ
Smooth (C∞)	all higher derivatives exist and are continuous	itself C∞ — infinitely differentiable	polynomials, eˣ, sin x, cos x

Summary: How f and f' Correspond

The two graphical translations covered above—reading

f'

from

f

, and reconstructing

f

from

f'

—are two views of a single dictionary connecting features of one function to features of the other. The table below collects this correspondence in compact form, organized by behavior of

f

with the parallel behavior of

f'

alongside; reading the columns in reverse order recovers the

f' \to f

mapping. Together with the vertical-translation ambiguity in the final row, this is everything the derivative function knows about the shape of

f

Behavior of f	Corresponding behavior of f'	Note
f is increasing	f'(x) > 0	graph of f' lies above the x-axis
f is decreasing	f'(x) < 0	graph of f' lies below the x-axis
f has a local extremum at a	f'(a) = 0, or f' undefined at a	horizontal tangent on f
f has a local maximum	f' changes sign + → −	f' crosses zero from above
f has a local minimum	f' changes sign − → +	f' crosses zero from below
f is steep at a	\|f'(a)\| is large	f' graph far from the x-axis
f is nearly flat at a	f'(a) ≈ 0	f' graph near the x-axis
f is linear on an interval	f' is constant on that interval	horizontal segment on the graph of f'
f is concave up	f' is increasing	slope of f is rising
f is concave down	f' is decreasing	slope of f is falling
f has an inflection point at a	f' has a local extremum at a	concavity of f flips
f shifted vertically by C	f' is unchanged	vertical translation is invisible to the derivative