Derivatives as Functions

The Derivative Function Defined

Computing the Derivative Function from the Definition

Derivative Notation: Function vs Point

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

Common Pitfalls

Derivative Function FAQ

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 →

The Derivative Function Defined

Fix a function

f

and consider the limit

Derivative Limit Definition

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}

Learn more about this formula: Derivative Limit Definition →

Browse all derivative formulas →

evaluated not at a single point

a

but at a variable

x

. Wherever this limit produces a finite value,

f'(x)

is defined. The result is a function

f'

whose input is

x

and whose output is the slope of

f

x

COROLLARY

The domain of

f'

is a subset of the domain of

f

Every point where

f

is differentiable belongs to the domain of

f'

; every point where the limit fails—corners, cusps, vertical tangents, discontinuities—is excluded. For a polynomial, the domains of

f

and

f'

coincide. For

f(x) = |x|

, the domain of

f

is all reals, but

x = 0

is missing from the domain of

f'

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

Computing the Derivative Function from the Definition

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

Pitfall: Squaring a sum term by term

Writing

(x + h)^2

x^2 + h^2

leaves

\frac{h^2}{h} = h

, which tends to

0

, so

f'(x) = 0

for every

x

. With

(x + h)^2 = x^2 + 2xh + h^2

the quotient is

2x + h

, and

f'(x) = 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.

Derivative Notation: Function vs Point

Notation

Derivative Notation: Function vs Point

The prime mark, the quotient it is built from, and the two ways to say “now evaluate at

a

” — the distinction this whole page turns on.

\lim

and its arrow — two-sided limits; the four systems for higher orders — higher-order derivatives; the quotient reading of

\frac{dy}{dx}

— differentials.

\frac{f(x+h) - f(x)}{h}

The difference quotient

Average slope over the step from

x

x+h

— the object whose limit is the derivative.

h

is a free increment, the same role

\Delta x

plays on differentials.

CasesThe

h

-form for computation; the

t

-form

\frac{f(t)-f(x)}{t-x}

in proofs, where the moving point has a name. Numerical work adds the symmetric form

\frac{f(x+h)-f(x-h)}{2h}

, accurate to one order higher.

Also written

\frac{\Delta y}{\Delta x}

— the increment form, standard in physics-flavoured texts.

Do not confuseA fraction you can evaluate at

h = 0

. There it is the indeterminate form

0/0

— the limit is the entire content of the definition.

f'(x)

f prime of x

One mark turns a function into its slope function. Binding order matters:

f'(x)

(f')(x)

— differentiate first, then feed in

x

Cases

f'(a)

— the same function evaluated at a point: one number, one slope. The domain shrinks where differentiability fails —

f(x) = |x|

loses

x = 0

on the way from

f

f'

Also written

\frac{dy}{dx}

\dot{y}

Df

— Leibniz, Newton, Euler; all four systems compared on higher-order derivatives.

Do not confuse“

f'(a)

= differentiate

f(a)

.” Substituting first leaves a constant, and its derivative is

0

— evaluation always comes after the prime.

Same glyph elsewhereThe prime marks a set complement

A'

and a transformed point in geometry — labels, not rates.

\left.\frac{dy}{dx}\right|_{x=a}

d y by d x, evaluated at x equals a

The evaluation bar. Prime notation evaluates by ordinary substitution —

f'(a)

— but

\frac{dy}{dx}

names no input slot, so Leibniz notation needs an explicit “now substitute” mark. The LaTeX reference builds it with \left.\frac{dy}{dx}\right|_{x=a}.

CasesAttaches to any operator expression, not only

\frac{dy}{dx}

. When the function has a name,

f'(a)

says the same thing shorter — the bar earns its keep when the derivative exists only as an expression.

Also written

\frac{dy}{dx}(a)

in some texts — compact but easy to misread as multiplication.

Do not confuseAn absolute value or a divider. The bar is an instruction, not an operation — and it must sit outside a finished derivative; substituting

a

before differentiating repeats the constant-trap above.

All calculus symbols →↑ Derivatives

f(x) = x³ above, f′(x) = 3x² below

Wherever the cubic rises its derivative sits above the axis, and at x = 0, where the cubic flattens for an instant, the derivative touches zero without crossing: the graph of f′ is read off the slopes of f. At the marker, x = 1, the cubic has slope 3 and the parabola has height 3. Slide the marker and watch height below follow slope above on the derivative explorer.

The reverse reading, from f′ back to f, uses the same dictionary in the other direction.

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' \geq 0

—the graph of

f'

lies on or above the

x

-axis (as for

x^3

x = 0

). Where

f

is decreasing,

f' \leq 0

—the graph of

f'

lies on or below the

x

-axis. Where

f

has a local extremum,

f'

crosses zero (or is undefined).

Pitfall: Reading the slope from the height

Taking the sign of

f'

from the height of

f

f(x) = x^2

is positive at

x = -1

, which suggests

f'(-1) > 0

. But

f

is falling there, and

f'(-1) = 2(-1) = -2

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'

Each feature of

f'

reports on the slope of

f

: sign gives direction, size gives steepness, trend gives bending.

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.

Pitfall: Taking every zero of the derivative as an extremum

Reading

f'(a) = 0

alone as a maximum or minimum fails for

f(x) = x^3

f'(0) = 0

, yet

f

rises on both sides of

0

. Since

f'(x) = 3x^2

does not change sign at

0

, there is no extremum there.

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

RULE

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.

One derivative, many functions

The section’s x² and x² + 5 idea with three curves: x² + 3, x² and x² − 2. Each is the others moved up or down, so at any x they all have the same slope. At x = 1.5 the three dashed tangents are parallel, each with slope 3, and everywhere the derivative is 2x. Reading f from f′ recovers this shape, but which of the curves is f needs one known value.

Knowing f(0) = 3 would single out x² + 3 from the whole family.

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:

Fact: Positive slope, yet not rising

For

g(x) = \frac{x}{2} + x^2 \sin(1/x)

with

g(0) = 0

, the slope at

0

g'(0) = \frac{1}{2}

, yet

g

is not increasing on any interval around

0

. Near

0

its derivative keeps swinging between about

-\frac{1}{2}

and

\frac{3}{2}

, so

g

falls again and again.

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.

THEOREMDarboux's theorem

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.

Fact: A derivative integration cannot undo

In 1881 Vito Volterra glued copies of

x^2 \sin(1/x)

onto a Cantor-like set of positive length. His function is differentiable everywhere with a bounded derivative, yet that derivative is discontinuous on too large a set to have a Riemann integral.

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.

Discontinuities of

f'

matter again when the derivative is integrated. Integration is meant to undo differentiation: integrating

f'

from

a

b

should give back

f(b) - f(a)

. When

f'

is continuous this always works, and it still works for

x^2 \sin(1/x)

, whose derivative stays bounded near

0

and misbehaves at only one point.

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

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Before you read

How smooth is the derivative?

Functions sort into classes by how well their derivatives behave. Each class sits inside the one around it.

C∞⊂C¹⊂D⊂C⁰

C⁰

Continuous

f'

may not exist

|x|

Differentiable

f'

exists everywhere but may be discontinuous

x² sin(1/x), f(0) = 0

C¹

Continuously differentiable

f'

exists and is continuous

polynomials, sin x, eˣ

C∞

Smooth

all higher derivatives exist and are continuous

Takeaway

Differentiable is not the same as C¹

A derivative can exist everywhere and still be discontinuous, as

x^2 \sin(1/x)

shows; by Darboux's theorem it can never have a jump.

Classes of functions by their derivative·/calculus/derivatives/functionLearn Math Class

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 on or above the x-axis
f is decreasing	f'(x) ≤ 0	graph of f' lies on or below the x-axis
f has a local extremum at a	f'(a) = 0, or f' undefined at a	horizontal tangent on f when f'(a) = 0 (corner or cusp when f' is undefined)
f has a local maximum	f' changes sign + → − (sufficient, not necessary)	f' crosses zero from above, or jumps from + to − at a corner or cusp
f has a local minimum	f' changes sign − → + (sufficient, not necessary)	f' crosses zero from below, or jumps from − to + at a corner or cusp
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

Common Pitfalls

Three mistakes cause most wrong answers when the derivative function is computed from the definition or read from a graph. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Squaring a sum term by term

In the limit definition for

f(x) = x^2

, the term

f(x + h) = (x + h)^2

is expanded as if a square could be taken one term at a time, and the cross term

2xh

is lost.

Wrong working:

\dfrac{(x + h)^2 - x^2}{h} = \dfrac{x^2 + h^2 - x^2}{h} = \dfrac{h^2}{h} = h

, and as

h \to 0

this gives

f'(x) = 0

for every

x

.

Right working:

(x + h)^2 = x^2 + 2xh + h^2

, so

\dfrac{(x + h)^2 - x^2}{h} = \dfrac{2xh + h^2}{h} = 2x + h

, which tends to

2x

. So

f'(x) = 2x

.

Why it happens: multiplication spreads over a sum,

3(x + h) = 3x + 3h

, and squaring is treated as if it did the same. It does not:

(x + h)^2 = (x + h)(x + h)

, and multiplying out produces the cross term

2xh

.

How to check: a derivative equal to

0

everywhere belongs to a flat graph, and the parabola is not flat. Numerically, at

x = 3

with

h = 0.01

the quotient is

\dfrac{3.01^2 - 9}{0.01} = 6.01

, close to

6

and far from

0

.

Reading the slope from the height

When the graph of

f'

is sketched from the graph of

f

, the sign of

f'

is taken from where

f

sits, above or below the

x

-axis, instead of from whether

f

is rising or falling.

Wrong working:

f(x) = x^2

lies above the

x

-axis for every

x \neq 0

, so

f'

is drawn positive everywhere, and

f'(-1) > 0

.

Right working: for

x < 0

the parabola falls, so

f'

is negative there. With

f'(x) = 2x

f'(-1) = -2

; the graph of

f'

is below the axis to the left of

0

and above it to the right.

Why it happens: both graphs are drawn on the same kind of axes, and copying the shape of

f

is easier than reading its slope point by point.

How to check: follow the tangent at a sample point. At

x = -1

the tangent to

x^2

goes down to the right, so its slope, which is the value of

f'

there, is negative.

Taking every zero of the derivative as an extremum

When

f

is reconstructed from

f'

, every point where

f'

equals

0

is marked as a maximum or a minimum, without checking whether

f'

changes sign there.

Wrong working: for

f(x) = x^3

f'(x) = 3x^2

0

x = 0

, so

f

is said to have a local extremum at

0

.

Right working:

3x^2 > 0

on both sides of

0

, so

f

rises before

0

and after it. The tangent at

0

is horizontal, but there is no extremum:

f(-0.1) = -0.001

is below

f(0) = 0

, and

f(0.1) = 0.001

is above it.

Why it happens: at a smooth maximum or minimum the tangent is always horizontal, and the statement is read backwards: horizontal tangent, therefore extremum.

How to check: look at the sign of

f'

just left and just right of the zero. An extremum needs a change of sign, from positive to negative for a maximum and from negative to positive for a minimum.

Derivative Function FAQ

What does the vertical bar in dy/dx|ₓ₌ₐ mean?

It means evaluate the derivative at x = a, after differentiating. Prime notation has an input slot, so f′(a) substitutes directly, but dy/dx names no slot — the bar supplies the missing “now substitute” instruction. It is not an absolute value or a divider, and it must sit outside a finished derivative: substituting a first leaves a constant, whose derivative is 0.Read more →

What is the difference between f′(a) and the derivative of f(a)?

f′(a) means differentiate f first, then substitute a — the prime binds to the function, not to the value. The derivative of f(a) is something else entirely: f(a) is a fixed number, and the derivative of a constant is 0. Evaluation always comes after differentiation, which is why the prime is attached to f rather than to f(a).Read more →

What is the symmetric difference quotient?

It is the two-sided version of the difference quotient, [f(x+h) − f(x−h)]/(2h), which steps forward and backward by h instead of forward only. Its limit is the same derivative, but for a fixed h it approximates the slope one order more accurately, so numerical differentiation almost always uses it. It can also return a value where no derivative exists.Read more →

Can a derivative have a jump discontinuity?

No. Darboux's theorem says every derivative has the intermediate value property: if f′ takes two values on an interval, it takes every value between them, and a jump would skip values. Derivatives can still be discontinuous — x²sin(1/x) is differentiable everywhere yet its derivative oscillates at 0 — but only oscillatory discontinuities are possible.Read more →