Higher Order Derivatives: Second, Third & nth

Definition

You have already taken the derivative of a derivative.

Differentiate

f(x) = x^3

and you get

f'(x) = 3x^2

, the slope of the curve at each point. Ask how fast that slope is changing and there is only one way to answer: differentiate

3x^2

, which gives

6x

. Nothing new was needed — the same operation, applied to its own result.

The solid curve is

f(x) = x^3

, the dashed one its derivative

3x^2

; the straight line is the tangent at one point, and its slope is the dashed curve's height there. Differentiating the dashed curve in turn gives

6x

. The same state is on the derivative visualizer.

The same move turns up in three different places.

In motion. If a position is

s(t) = t^3

, its derivative is the velocity

3t^2

, and the derivative of the velocity is the acceleration

6t

.

time t	position t³	velocity 3t²	acceleration 6t
0	0	0	0
1	1	3	6
2	8	12	12
3	27	27	18

In the shape of a curve. The slope of

y = x^3

is

3x^2

; whether that slope is shrinking or growing — whether the curve bends down or up — is the sign of its derivative,

6x

.

In plain algebra. Keep differentiating

x^3

and the powers step down until nothing is left.

start	once	twice	three times	four times
x³	3x²	6x	6	0

Left of

0

,

6x < 0

: the slope shrinks and the curve bends down. Right of

0

,

6x > 0

: the slope grows and the curve bends up. The dot at

0

is where

6x

changes sign.

What the three share is a rate of change of a rate of change.

Acceleration is how fast velocity changes; bending is how fast slope changes; each column of the chain is how fast the previous column changes. In every case a derivative is taken, and then the derivative of that result. The surface is different — seconds and metres, curves, powers of

x

— and the operation underneath is the same.

The definition

The second derivative of

f

is the derivative of

f'

. Each higher-order derivative is the derivative of the one before: the third is the derivative of

f''

, and the

n

th,

f^{(n)}

, is

f

differentiated

n

times.

Definition

f''(x) = \frac{d}{dx}[f'(x)]

For

f(x) = x^3

:

f'' = 6x

,

f''' = 6

, and every derivative after that is

0

.

A higher-order derivative is a function, just as the first derivative is.

f''(x) = 6x

is a new function with its own graph; evaluated at a point it gives a number,

f''(2) = 12

. It is not a different kind of object from

f'

— only one step further down the same chain.

What the name buys: each fact about it is proved once and used everywhere.

Once "second derivative" is a thing with a name and a symbol, one result serves motion, curves and series at once.

once it has a name	what it settles
the sign of f″	which way a curve bends, and where an inflection is
f″ at a critical point	minimum or maximum, by the second derivative test
s″(t)	acceleration, for any motion
every f⁽ⁿ⁾(a)	the coefficients of a Taylor series

The chain also has a limit of its own.

COROLLARY

Each order depends on the one below it:

f^{(n)}

exists at a point only if

f^{(n-1)}

exists in a neighbourhood of it and is differentiable there.

The chain fails quickly for functions with corners, and the order at which it fails is what the smoothness classes below measure.

The rest of the page covers the notation for every order, the second derivative and its physical meaning, the patterns that repeated differentiation produces for polynomials, exponentials, sine and cosine, nth-derivative formulas, the link to Taylor series, and smoothness classes.

Notation for Higher-Order Derivatives

Notation

Notation for Higher-Order Derivatives

Four systems, all still in use. Which one a text picks depends on whether the variable of differentiation matters, whether it is time, and whether derivatives are being manipulated as objects.

f'

and

\frac{dy}{dx}

are defined on Derivatives. Everything below extends them past order one.

f'

·

f''

·

f^{(n)}

f prime, f double prime, f superscript n in parentheses

Lagrange's system. Primes count the order up to three; past that they stop being countable and a parenthetical superscript takes over:

f^{(4)}

,

f^{(n)}

. Compact, and silent about the variable — which is why it fails as soon as more than one is in play.

Cases

f^{(0)} = f

— the zeroth derivative is the function itself, which is what lets a Taylor series start its sum at

n = 0

. Older European texts write low orders as roman numerals:

f^{iv}

,

f^{v}

.

Also written

y'

,

y''

when the function is named by its output rather than by

f

.

f''''

turns up for the fourth, but rarely survives an editor.

Do not confuseThe parentheses are the whole point.

f^n

is a power,

f^{\circ n}

is composition with itself,

f^{-1}

is the inverse. Four meanings compete for one superscript slot, and only the derivative brackets itself.

Same glyph elsewhereThe prime marks a transformed or corresponding object almost everywhere else —

A'

for a mapped point in geometry,

x'

for a transformed coordinate. Nothing to do with rates.

\frac{d^n y}{dx^n}

d to the n y, d x to the n

Leibniz's system, extended. Reach for it when the variable of differentiation matters, or when the differential itself is in play. The whole symbol is one operator applied to

y

— not a fraction with

d^n y

over

dx^n

.

CasesSecond derivative:

\frac{d^2 y}{dx^2}

, read "d two y, d x squared", meaning

\frac{d}{dx}\!\left(\frac{dy}{dx}\right)

. The operator form

\frac{d^n}{dx^n}f

separates the instruction from the target, which is what makes it usable inside differentiation rules.

Also written

\partial^n y / \partial x^n

once other variables are held fixed; the keyboard carries

\partial

.

Do not confuseThe exponent sits in two different places: above the

d

in the numerator, beside the

x

in the denominator. Not a typographical accident — it records that

d

is applied

n

times while

x

is the thing being squared, cubed, and so on in the dimensional bookkeeping.

\dot{y}

·

\ddot{y}

y dot, y double dot

Newton's notation. Each dot is one derivative with respect to time — never space. Standard in mechanics and differential equations, where the independent variable is always

t

.

CasesOne dot, velocity. Two, acceleration. Three is the practical ceiling; beyond that the dots stop being countable and texts revert to

y^{(n)}

.

Also written

\dot{y} = \frac{dy}{dt}

, the Leibniz form it abbreviates.

Do not confuseA dot is not a generic derivative mark.

\dot{x}

where

x

varies over space is wrong, not merely unusual — the notation carries its variable inside it.

D^n f

D to the n of f

Euler's operator form.

D

means differentiate; the exponent counts applications. Write

D_x^n f

when several variables are in play. Used wherever derivatives are handled algebraically rather than computed.

Cases

D^0 f = f

by convention, which is what makes

D

usable inside a sum or a polynomial in

D

— the basis of operator methods for linear differential equations.

Also written

\mathcal{D}

where plain

D

is needed for something else.

Do not confuse

\Delta

is the difference operator, not the derivative:

\Delta f = f(x+h) - f(x)

. The two sit side by side in numerical analysis and are separated by a single limit.

All calculus symbols →↑ Derivatives

Order	Lagrange notation	Leibniz notation	Read aloud
1	f'(x)	dy ⁄ dx	"f prime" ; "d y, d x"
2	f''(x)	d²y ⁄ dx²	"f double prime" ; "d two y, d x squared"
3	f'''(x)	d³y ⁄ dx³	"f triple prime"
4 and beyond	f⁽⁴⁾(x), f⁽⁵⁾(x)	d⁴y ⁄ dx⁴, d⁵y ⁄ dx⁵	parentheses distinguish derivative order from an exponent
n (general)	f⁽ⁿ⁾(x)	dⁿy ⁄ dxⁿ	the nth derivative — a single operator applied n times

f″(c) = 2.60: the slope is increasing

The first derivative says whether the curve rises; the second says whether the rise is speeding up. At the marker the tangent slopes are growing as x increases, the curve bends upward, and the readout f″(c) = 2.60 records how fast. Where the readout turns negative the bend has flipped. Watch the number and the bend change together on the concavity and inflection points tool.

The second derivative is the first one whose sign has a shape of its own on the graph.

The Second Derivative

The second derivative

f''(x)

measures the rate of change of the slope. Where

f''(x) > 0

, the slope

f'(x)

is increasing—the function bends upward (concave up). Where

f''(x) < 0

, the slope is decreasing—the function bends downward (concave down).

Pitfall: Squaring the first derivative

Reading

f''(x)

or

\frac{d^2y}{dx^2}

as the square of the first derivative turns

y = x^3

into

(3x^2)^2 = 9x^4

, which is never negative. Differentiating twice gives

f''(x) = 6x

, and

f''(-1) = -6

: the curve bends down there.

This information is independent of whether

f

itself is increasing or decreasing. A function can rise while decelerating (

f' > 0

,

f'' < 0

) or fall while accelerating in the negative direction (

f' < 0

,

f'' < 0

). The first and second derivatives describe different aspects of behavior.

The second derivative also powers the second derivative test for classifying critical points. At a point where

f'(c) = 0

: if

f''(c) > 0

, the critical point is a local minimum; if

f''(c) < 0

, a local maximum; if

f''(c) = 0

, the test is inconclusive.

Inflection points—where concavity reverses—occur where

f''

changes sign. When

f''(c)

exists, the condition

f''(c) = 0

is necessary but not sufficient (and

f''

may instead fail to exist, as for

x^{1/3}

at

0

); the sign of

f''

must actually switch across

c

.

The first derivative says which way a graph is heading; the second says which way it bends.

Physical Interpretation

In kinematics, the first three derivatives of position

s(t)

have standard names.

The first derivative

s'(t)

is velocity: the rate of change of position. It tells how fast an object moves and in which direction.

The second derivative

s''(t)

is acceleration: the rate of change of velocity. Positive acceleration means speeding up (in the positive direction) or decelerating (in the negative direction). The sign of acceleration relative to the sign of velocity determines whether the object is speeding up or slowing down.

Pitfall: Reading negative acceleration as slowing down

Taking

s''(t) < 0

to mean slowing down fails for

s(t) = -t^2

: at

t = 1

,

s''(1) = -2

, yet the speed

|s'(t)| = 2t

grows from

2

at

t = 1

to

4

at

t = 2

. Velocity

-2

and acceleration

-2

have the same sign, so the object is speeding up.

For the position

s(t) = t^3

from the first section, the velocity is

3t^2

and the acceleration is

6t

. At

t = 2

the velocity is

12

and the acceleration is

12

, both positive, and the table of values there shows the object covering more ground in each second than in the one before. Speed is the size of the velocity,

|s'(t)|

; it carries no direction, while velocity and acceleration both do.

The third derivative

s'''(t)

is jerk: the rate of change of acceleration. Jerk is felt physically as a sudden push or lurch—smooth motion has low jerk, while abrupt starts and stops produce high jerk. Elevator design, roller coaster engineering, and vehicle ride comfort all involve controlling jerk.

Fact: Why bends start with a spiral

Joining straight track directly to a circular bend would make the sideways acceleration jump from

0

to

\frac{v^2}{R}

in an instant, an infinite jerk. Railways and highways insert a spiral whose curvature grows in proportion to distance, so the acceleration builds at a steady rate.

Jerk also appears where the speed never changes. A train running at constant speed

v

along straight track has no acceleration at all. On a circular bend of radius

R

it has a sideways acceleration of

\frac{v^2}{R}

, directed toward the centre of the circle, which passengers feel as a push toward the outside of the bend. Somewhere between the straight and the bend that acceleration has to appear, and the rate at which it appears is a jerk.

Beyond the third derivative, the terms snap (

s^{(4)}

), crackle (

s^{(5)}

), and pop (

s^{(6)}

) are used in specialized engineering contexts but rarely appear in standard calculus. The table below collects all six in order, with each name paired with what the derivative measures.

Order	Derivative of position	Standard name	What it measures
1	s'(t)	velocity	rate of change of position; speed and direction of motion
2	s''(t)	acceleration	rate of change of velocity; force-related quantity in Newton's second law
3	s'''(t)	jerk	rate of change of acceleration; felt physically as a sudden push or lurch
4	s⁽⁴⁾(t)	snap (or jounce)	specialised engineering use; rarely appears in standard calculus
5	s⁽⁵⁾(t)	crackle	named by analogy with snap; specialised engineering use
6	s⁽⁶⁾(t)	pop	named by analogy with snap and crackle; specialised engineering use

Patterns in Repeated Differentiation — Polynomials

COROLLARY

Polynomials terminate under repeated differentiation. Each differentiation reduces the degree by one, so a polynomial of degree

n

reaches a constant after

n

differentiations and becomes zero after

n + 1

.

For

f(x) = x^5

:

Patterns in Repeated Differentiation — Polynomials

f'(x) = 5x^4, \quad f''(x) = 20x^3, \quad f'''(x) = 60x^2, \quad f^{(4)}(x) = 120x, \quad f^{(5)}(x) = 120, \quad f^{(6)}(x) = 0

The coefficient at the

n

th derivative of

x^n

is

n! = n(n-1)(n-2) \cdots 1

, accumulated from the power rule applied

n

times. Specifically,

\frac{d^n}{dx^n}[x^n] = n!

and

\frac{d^k}{dx^k}[x^n] = 0

for all

k > n

.

For a general polynomial

p(x) = a_n x^n + \cdots + a_1 x + a_0

, the

n

th derivative is

n! \cdot a_n

, a constant. This relationship becomes central in Taylor series, where the coefficient

a_n

is recovered as

\frac{f^{(n)}(a)}{n!}

.

Patterns in Repeated Differentiation — Exponentials

The natural exponential

e^x

is unchanged by differentiation:

nth Derivative of Natural Exponential

\frac{d^n}{dx^n}[e^x] = e^x

Learn more about this formula: nth Derivative of Natural Exponential →

Every derivative of

e^x

is

e^x

. The only functions with this property are the constant multiples

Ce^x

(including the trivial

f(x) = 0

).

For the general exponential

e^{ax}

, the chain rule introduces a factor of

a

at each step:

nth Derivative of Scaled Exponential

\frac{d^n}{dx^n}[e^{ax}] = a^n e^{ax}

Learn more about this formula: nth Derivative of Scaled Exponential →

Browse all derivative formulas →

Pitfall: Using the chain-rule factor only once

Writing

\frac{d^3}{dx^3}[e^{2x}] = 2e^{2x}

keeps the factor

2

from the first step only. Differentiating three times gives

4e^{2x}

after the second step and

8e^{2x}

after the third, so the answer is

8e^{2x}

.

Each differentiation multiplies by

a

. After

n

differentiations, the accumulated constant is

a^n

. This pattern appears in solutions to differential equations, where

e^{ax}

satisfies equations whose characteristic root is

a

.

For

a^x

with arbitrary base: since

a^x = e^{x \ln a}

, the

n

th derivative is

(\ln a)^n \cdot a^x

. The factor

\ln a

replaces

a

in the exponential pattern.

Patterns in Repeated Differentiation — Sine and Cosine

The derivatives of

\sin x

cycle with period four:

Patterns in Repeated Differentiation — Sine and Cosine

\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x \to \cdots

The

n

th derivative of

\sin x

depends on

n \mod 4

:

Patterns in Repeated Differentiation — Sine and Cosine

\frac{d^n}{dx^n}[\sin x] = \begin{cases} \sin x & n \equiv 0 \pmod{4} \\ \cos x & n \equiv 1 \pmod{4} \\ -\sin x & n \equiv 2 \pmod{4} \\ -\cos x & n \equiv 3 \pmod{4} \end{cases}

A compact formula captures all four cases:

\frac{d^n}{dx^n}[\sin x] = \sin\left(x + \frac{n\pi}{2}\right)

. The same holds for cosine:

\frac{d^n}{dx^n}[\cos x] = \cos\left(x + \frac{n\pi}{2}\right)

.

nth Derivative of Sine

\frac{d^n}{dx^n}[\sin x] = \sin\!\left(x + \frac{n\pi}{2}\right)

Learn more about this formula: nth Derivative of Sine →

nth Derivative of Cosine

\frac{d^n}{dx^n}[\cos x] = \cos\!\left(x + \frac{n\pi}{2}\right)

Learn more about this formula: nth Derivative of Cosine →

Browse all derivative formulas →

Pitfall: Dropping the minus sign

Taking the derivative of

\cos x

to be

\sin x

makes the cycle

\sin x \to \cos x \to \sin x

, of length two, and gives

\frac{d^2}{dx^2}[\sin x] = \sin x

. With

(\cos x)' = -\sin x

the second derivative is

-\sin x

, and the cycle closes only after four steps.

For

\sin(ax)

, the chain rule introduces a factor of

a

per differentiation:

\frac{d^n}{dx^n}[\sin(ax)] = a^n \sin\left(ax + \frac{n\pi}{2}\right)

. The cycle in the trigonometric part persists; only the amplitude grows as

a^n

.

This four-fold periodicity distinguishes trigonometric derivatives from polynomial derivatives (which terminate) and exponential derivatives (which replicate).

Differentiating sine is a shift

The curves sin x and its derivative cos x are the same wave, one a quarter period ahead of the other: the peak of sin x at x = π/2 sits at x = 0 on cos x. That is (sin x)′ = sin(x + π/2). Each further derivative moves the wave another π/2 to the left, so four of them move it a full period 2π and bring back sin x, which is the four-step cycle sin, cos, −sin, −cos.

For sin(ax) the shift is the same; only the height is multiplied by a at each step.

The nth Derivative of Specific Forms

Several standard functions have known closed-form

n

th derivatives.

For

f(x) = \frac{1}{x}

: rewriting as

x^{-1}

and applying the power rule repeatedly gives

nth Derivative of Reciprocal

\frac{d^n}{dx^n}\!\left[\frac{1}{x}\right] = \frac{(-1)^n\, n!}{x^{n+1}}

Learn more about this formula: nth Derivative of Reciprocal →

Each differentiation multiplies by one more negative integer, producing the factorial and the alternating sign.

For

f(x) = \ln x

: since

f'(x) = x^{-1}

, the higher derivatives follow the pattern above shifted by one:

nth Derivative of Natural Logarithm

\frac{d^n}{dx^n}[\ln x] = \frac{(-1)^{n-1}\, (n-1)!}{x^n}

Learn more about this formula: nth Derivative of Natural Logarithm →

For

f(x) = x^m

where

m

is a positive integer and

n \leq m

:

nth Derivative of Power

\frac{d^n}{dx^n}[x^m] = \frac{m!}{(m-n)!}\, x^{m-n} \quad (n \leq m)

Learn more about this formula: nth Derivative of Power →

Browse all derivative formulas →

The coefficient

\frac{m!}{(m-n)!}

is the falling factorial, counting the multipliers accumulated over

n

applications of the power rule.

Fact: Half a derivative brings in

\pi

With factorials extended to fractions,

\left(\frac{1}{2}\right)! = \frac{\sqrt{\pi}}{2}

, and the formula gives

\frac{d^{1/2}}{dx^{1/2}}[x] = \frac{2\sqrt{x}}{\sqrt{\pi}}

. Applying the same half-derivative once more gives exactly

1

, the ordinary derivative of

x

.

Once the pattern is written as a single formula, the order

n

looks like any other number in it. The question of what a derivative of order

\frac{1}{2}

would mean was asked almost as soon as the notation existed: in 1695 L'Hôpital put it to Leibniz in a letter, and Leibniz answered that it led to a paradox from which useful consequences would one day be drawn. An answer needs factorials of numbers that are not whole, and the gamma function supplies them.

These closed-form expressions are useful for computing specific high-order derivatives without performing each differentiation step individually.

Higher-Order Derivatives and Taylor Series

The Taylor series of

f

centered at

x = a

is

Taylor Series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n

Learn more about this formula: Taylor Series →

Browse all derivative formulas →

Pitfall: Leaving out the factorials

Using

f^{(n)}(a)

as the coefficient without dividing by

n!

turns the cubic Taylor polynomial of

\sin x

at

0

into

x - x^3

, which gives

0.375

at

x = 0.5

. With

\frac{f'''(0)}{3!} = -\frac{1}{6}

it is

x - \frac{x^3}{6} \approx 0.47917

, close to

\sin 0.5 \approx 0.47943

.

Each coefficient depends on a higher-order derivative evaluated at the center point

a

. The zeroth derivative

f^{(0)}(a) = f(a)

gives the constant term. The first derivative gives the linear term. The second derivative gives the quadratic correction. Each successive term captures finer detail about how

f

deviates from the lower-order approximation.

The Taylor polynomial of degree

k

truncates the series after

k + 1

terms, providing a polynomial approximation to

f

near

a

. For well-behaved functions such as

\sin x

, the approximation improves with

k

—more derivatives typically mean a closer fit near

a

, often over a wider interval.

The connection between higher-order derivatives and Taylor series gives these derivatives their deepest significance: the complete collection

\{f^{(n)}(a)\}_{n=0}^{\infty}

determines

f

locally (for analytic functions). For such functions, knowing all derivatives at a single point reconstructs the entire function in a neighborhood of that point.

Taylor polynomials of sin x at 0

The Taylor polynomials of sin x at 0 use the derivatives 0, 1, 0, −1, 0, 1, … there. Degree 1 is the line y = x, which follows the curve only near 0. Degree 3, x − x³/6, stays close until about ±1.5; degree 5 adds x⁵/120 and holds on to about ±2.5. Each additional derivative used at the single point 0 widens the stretch where the polynomial matches sin x.

Only odd powers appear because the even-order derivatives of sin x at 0 are all 0.

For analytic functions, the derivatives at one point rebuild the whole function around it.

Existence and Smoothness Classes

A function may be differentiable once but not twice. The function

f(x) = x|x|

has

f'(x) = 2|x|

, which is continuous but not differentiable at

x = 0

. So

f

is in class

C^1

(continuously differentiable) but not

C^2

.

The smoothness classes organize functions by how many continuous derivatives they possess. A function belongs to

C^n

if

f, f', f'', \ldots, f^{(n)}

all exist and are continuous. Class

C^0

is simply continuous functions. Class

C^\infty

consists of infinitely differentiable functions—called smooth functions—where derivatives of all orders exist and are continuous.

Polynomials,

e^x

,

\sin x

,

\cos x

, and their compositions are all

C^\infty

. Piecewise functions typically belong to a finite

C^n

class determined by how smoothly the pieces join at their boundaries: matching values gives

C^0

, matching first derivatives gives

C^1

, and so on.

There exist functions that are

C^\infty

but not analytic—their Taylor series either diverges or converges to something other than the function itself. The standard example is

f(x) = e^{-1/x^2}

for

x \neq 0

and

f(0) = 0

: every derivative at

x = 0

is zero, so the Taylor series is identically zero, yet the function is not zero away from the origin.

THEOREM

Smooth does not automatically mean representable by a power series.

The table below lays out the main tiers — C⁰, the general Cⁿ, C∞, and the strictly stronger analytic class — alongside what each requires and characteristic examples.

Smooth to the eye, not twice differentiable

The section’s f(x) = x|x| bends smoothly through the origin; it is x² on the right and −x² on the left. Its derivative f′(x) = 2|x| is continuous, but it is a V with a corner at 0, and a corner has no derivative. So f′′(0) does not exist: f is in C¹ but not in C².

Joining x² and −x² matches values and first derivatives at 0, but not second derivatives, 2 and −2.

Class	What it requires	Examples
C⁰	f is continuous (no derivatives required)	\|x\|; piecewise functions whose pieces match values at the boundary
Cⁿ	f, f', f'', … , f⁽ⁿ⁾ all exist and are continuous	x\|x\| belongs to C¹ but not C²
C^∞ (smooth)	derivatives of all orders exist and are continuous	polynomials, e^x, sin x, cos x, and their compositions
Analytic	C^∞, and the Taylor series at every point converges to f near that point	most elementary functions; counterexample f(x) = e^{−1 ⁄ x²} (extended by f(0) = 0) is C^∞ but not analytic at 0

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Before you read

Classes by number of continuous derivatives

A function is in

C^n

when

f, f', \ldots, f^{(n)}

all exist and are continuous. Each level up asks for more, so every class sits inside the one below it.

strongestsmoothnessweakest

Analyticequal to its Taylor series

C^\infty

— smoothderivatives of all orders: polynomials,

e^x

,

\sin x

C^n

n

continuous derivatives:

x|x|

is

C^1

, not

C^2

C^0

continuous functions

Takeaway

Smooth is not the top

Analytic is strictly stronger than

C^\infty

:

e^{-1/x^2}

(with

f(0) = 0

) is smooth, yet its Taylor series at

0

is identically zero and does not represent it.

Smoothness classes·/calculus/derivatives/higher-orderLearn Math Class

nth Derivatives at a Glance

Different function families behave very differently under repeated differentiation: polynomials terminate, exponentials replicate, and trigonometric functions cycle. The table below collects the closed-form nth derivative for each standard function discussed above into a single reference card, with a brief note on what drives the pattern.

Function	nth derivative	Note
x^m (m a positive integer)	m! ⁄ (m − n)! · x^{m − n} for n ≤ m	falling factorial; becomes the constant m! when n = m, and zero for n > m
Polynomial of degree m	m! · a_m (when n = m); 0 (when n > m)	terminates after m + 1 differentiations
1 ⁄ x	(−1)ⁿ · n! ⁄ x^{n + 1}	alternating sign accompanied by factorial growth in the numerator
ln x	(−1)^{n − 1} · (n − 1)! ⁄ xⁿ (n ≥ 1)	the 1 ⁄ x pattern shifted by one order, since (ln x)' = 1 ⁄ x
e^x	e^x	self-replicating; together with its constant multiples Ce^x, the only functions with this property
e^{a x}	aⁿ · e^{a x}	each differentiation multiplies by a, so the constant accumulates as aⁿ
a^x	(ln a)ⁿ · a^x	since a^x = e^{x · ln a}, ln a takes the role of the multiplier
sin x	sin(x + n π ⁄ 2)	four-fold cycle: sin → cos → −sin → −cos → sin
cos x	cos(x + n π ⁄ 2)	same four-fold cycle, starting from cosine
sin(a x)	aⁿ · sin(a x + n π ⁄ 2)	trigonometric cycle persists; amplitude grows as aⁿ

Common Pitfalls

Five mistakes cause many wrong answers with higher-order derivatives. Each is marked by a box 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 the first derivative

The second derivative is computed as the square of the first:

f''(x)

is read as

(f'(x))^2

, or

\frac{d^2y}{dx^2}

as

\left(\frac{dy}{dx}\right)^2

.

Wrong working: for

y = x^3

,

\frac{dy}{dx} = 3x^2

, so

\frac{d^2y}{dx^2} = (3x^2)^2 = 9x^4

. This is never negative, so the curve would be concave up everywhere.

Right working: differentiate

3x^2

once more,

\frac{d^2y}{dx^2} = 6x

. At

x = -1

it is

-6

, so the curve bends down to the left of

0

and up to the right, as the first section shows.

Why it happens: in

\frac{d^2y}{dx^2}

the

2

looks like an exponent, and in

f''

the second prime looks like a repeat of the first; both suggest a product of two first derivatives.

How to check: compare the degrees. Each differentiation lowers the degree of a polynomial by one, so the second derivative of

x^3

must have degree

1

, not

4

.

Reading negative acceleration as slowing down

The sign of

s''(t)

alone is taken to decide whether an object speeds up or slows down: a negative acceleration is read as braking, a positive one as speeding up.

Wrong working: for

s(t) = -t^2

,

s''(t) = -2 < 0

, so the object is slowing down.

Right working: the velocity is

s'(t) = -2t

, and for

t > 0

the speed is

|s'(t)| = 2t

. At

t = 1

the speed is

2

, at

t = 2

it is

4

: the object speeds up. Velocity and acceleration are both negative, and when they have the same sign the speed grows.

Why it happens: in everyday speech "acceleration" means speeding up and "deceleration" slowing down. In calculus a negative acceleration only says that the velocity decreases, and a negative velocity that decreases gets larger in size.

How to check: compare the signs of

s'(t)

and

s''(t)

at the moment in question, or compute the speed

|s'(t)|

at two nearby times.

Using the chain-rule factor only once

When

e^{ax}

is differentiated several times, the factor

a

from the chain rule is written once, as if it appeared only at the first step.

Wrong working:

\frac{d^3}{dx^3}[e^{2x}] = 2e^{2x}

.

Right working: each step applies the chain rule again.

\frac{d}{dx}[e^{2x}] = 2e^{2x}

,

\frac{d^2}{dx^2}[e^{2x}] = 4e^{2x}

,

\frac{d^3}{dx^3}[e^{2x}] = 8e^{2x}

, which is

2^3 e^{2x}

, as the formula

a^n e^{ax}

says.

Why it happens: after the first step,

2e^{2x}

looks like a constant times an exponential, and the exponential is then treated like

e^x

, whose derivative is itself.

How to check: count the factors. Three differentiations bring in three chain-rule factors of

2

, so the coefficient must be

2 \cdot 2 \cdot 2 = 8

. The same count for

a^x

gives

(\ln a)^n

.

Dropping the minus sign

The derivative of

\cos x

is taken to be

\sin x

instead of

-\sin x

. The error is carried into every later step, and the cycle comes out with the wrong length.

Wrong working:

\sin x \to \cos x \to \sin x \to \cos x

, so

\frac{d^2}{dx^2}[\sin x] = \sin x

and

\frac{d^3}{dx^3}[\sin x] = \cos x

.

Right working:

\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x

. So

\frac{d^2}{dx^2}[\sin x] = -\sin x

,

\frac{d^3}{dx^3}[\sin x] = -\cos x

, and the cycle closes after four steps.

Why it happens: the derivative of

\sin x

has no minus sign, and the derivative of

\cos x

is remembered by symmetry with it.

How to check: use the compact formula

\sin\left(x + \frac{n\pi}{2}\right)

; for

n = 2

it gives

\sin(x + \pi) = -\sin x

. Or test one point:

\sin x

has a maximum at

x = \frac{\pi}{2}

, so its second derivative cannot be positive there, and

-\sin\frac{\pi}{2} = -1

is negative.

Leaving out the factorials

In the Taylor series, the coefficient of

(x - a)^n

is taken to be

f^{(n)}(a)

, without the division by

n!

.

Wrong working: for

\sin x

at

a = 0

, the values

f(0), f'(0), f''(0), f'''(0)

are

0, 1, 0, -1

, so the cubic Taylor polynomial is

x - x^3

. At

x = 0.5

it gives

0.375

.

Right working: divide each derivative by

n!

. The cubic term is

\frac{-1}{3!}x^3 = -\frac{x^3}{6}

, and the polynomial is

x - \frac{x^3}{6}

. At

x = 0.5

it gives

0.5 - 0.02083 \approx 0.47917

, while

\sin 0.5 \approx 0.47943

.

Why it happens: the first two terms,

f(a)

and

f'(a)(x - a)

, show no division, because

0! = 1! = 1

, and the pattern seems to continue without it.

How to check: differentiate the polynomial

n

times at the center; it must give back

f^{(n)}(a)

. The third derivative of

x - x^3

is

-6

, not

f'''(0) = -1

; the third derivative of

x - \frac{x^3}{6}

is

-1

.

Higher-Order Derivatives FAQ

Why is the nth derivative written f⁽ⁿ⁾ with parentheses?

+

Because the superscript slot is already crowded. Without parentheses, fⁿ reads as a power, f∘ⁿ is n-fold composition, and f⁻¹ is the inverse — three other meanings competing for the same position. The parentheses mark the number as a derivative count and nothing else. Primes handle orders one through three; past that they stop being countable.Read more →

Why is the second derivative written d²y/dx² and not d²y/d²x?

+

Because the two exponents count different things. In the numerator, d² records that the operator d has been applied twice to y. In the denominator, dx² is really (dx)² — the increment dx squared, from the dimensional bookkeeping. The whole symbol is one operator applied to y, not a fraction of d²y over dx².Read more →

What is jerk in calculus?

+

Jerk is the third derivative of position with respect to time — the rate at which acceleration changes. You feel it as a sudden lurch rather than as speed or force, which is why elevator, roller-coaster, and vehicle design all work to keep it small. The fourth, fifth, and sixth derivatives are sometimes called snap, crackle, and pop.Read more →

What is the nth derivative of 1/x and ln x?

+

For 1/x the nth derivative is (−1)ⁿ·n!/x^(n+1); for ln x it is (−1)^(n−1)·(n−1)!/xⁿ. The two are the same pattern shifted by one step, because the first derivative of ln x is 1/x. Each differentiation multiplies by one more negative integer, which is where the factorial and the alternating sign both come from.Read more →

Is every smooth function equal to its Taylor series?

+

No. A function can have derivatives of every order and still not be analytic. The standard example is f(x) = e^(−1/x²) with f(0) = 0: every derivative at the origin is zero, so its Taylor series there is identically zero, yet the function is positive everywhere else. Smooth guarantees the series exists, not that it converges to f.Read more →

Higher Order Derivatives

Differentiating the Derivative

Key Terms

Definition

Notation for Higher-Order Derivatives

Notation for Higher-Order Derivatives

The Second Derivative

Physical Interpretation

Patterns in Repeated Differentiation — Polynomials

Patterns in Repeated Differentiation — Exponentials

Patterns in Repeated Differentiation — Sine and Cosine

The nth Derivative of Specific Forms

Higher-Order Derivatives and Taylor Series

Existence and Smoothness Classes

nth Derivatives at a Glance

Common Pitfalls

Higher-Order Derivatives FAQ

Why is the nth derivative written f⁽ⁿ⁾ with parentheses?

Why is the second derivative written d²y/dx² and not d²y/d²x?

What is jerk in calculus?

What is the nth derivative of 1/x and ln x?

Is every smooth function equal to its Taylor series?