Higher-order derivatives — notation, physical meaning

4.1.24 · Maths › Calculus I — Limits & Derivatives

Higher-order derivative KIYA hota hai?

KYUN zaroori hai? Kyunki pehli derivative sirf slope batati hai. Yeh jaanne ke liye ki wo slope tez ho rahi hai ya dheemi (curvature, acceleration, concavity), aapko second derivative chahiye — aur aise hi aage.

Notation (ek hi idea, kaafi costumes mein)

Leibniz $\frac{d ^{2} y}{d x ^{2}}$ KYUN likhta hai? Isko $(\frac{d}{d x})^{2} y$ ke roop mein samjho: operator $\frac{d}{d x}$ ko do baar apply karo. Upar ka " $2$ " batata hai kitne $d$ hain; neeche ka " $2$ " batata hai kitne $d x$ hain — yeh $y$ ya $x$ ki power nahi hai, yeh operator ko do baar apply karne ka hisaab hai.

Physical meaning — kinematic ladder

$s (t)$ ki Derivative	Naam	Matlab
$s$	position	aap kahan ho
$s^{'} = v$	velocity	position kitni tezi se change ho rahi hai
$s^{''} = a$	acceleration	velocity kitni tezi se change ho rahi hai
$s^{'''} = j$	jerk	acceleration kitni tezi se change ho rahi hai

Figure — Higher-order derivatives — notation, physical meaning

Worked Examples

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek car mein ho. Kahan ho road par — woh hai position. Kitni tezi se chal rahe ho — woh hai velocity. Jab tum gas dabate ho aur seat mein dhakka lagta hai, woh "tezi aane" ki feeling hai acceleration — yeh tumhari speed ka change hai. Ab agar tum achanak pedal itni zyada dabao ki woh dhakka khud uchhal jaaye, woh jolt hai jerk. Har ek bas "pehli cheez kitni tezi se change ho rahi hai?" hai. Maths mein bhi exactly yehi hota hai: ek derivative lete ho, phir us ki bhi derivative lete ho, baar baar. Aasaan!

Active Recall

The

n

-th derivative

f^{(n)}

is defined as

f

ko kul

n

baar differentiate karne ka result:

f^{(n)} = \frac{d}{d x} f^{(n - 1)}

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

means

\frac{d}{d x}

ko do baar apply karo — derivative ki derivative — NOT

(d y / d x)^{2}

Second time-derivative of position is called

acceleration,

a = \overset{s}{¨} = \frac{d ^{2} s}{d t ^{2}}

Third time-derivative of position is called

jerk,

j = s ... = \frac{d ^{3} s}{d t ^{3}}

f^{''} > 0

on an interval means the graph is

concave up (slope badh raha hai, cup shape

⌣

f^{''} < 0

means the graph is

concave down (slope ghad raha hai,

⌢

An inflection point occurs where

f^{''}

apna sign badalta hai (aksar

f^{''} = 0

par).

The

n

-th derivative of

sin x

equals

sin (x + \frac{nπ}{2})

— derivatives period 4 ke saath cycle karte hain.

For

f (x) = x^{4}

f^{(5)} (x) =

0

— degree-

n

polynomial

n + 1

derivatives ke baad zero ho jaata hai.

Why is

\frac{d ^{2} y}{d x ^{2}} \neq = (\frac{d y}{d x})^{2}

Superscripts operator applications count karte hain, powers nahi; jaise

x^{3}

ke liye yeh

6 x

9 x^{4}

dete hain.

Connections

Derivative — definition as a limit
Power Rule and differentiation rules
Concavity and the Second-Derivative Test
Inflection points
Kinematics — position, velocity, acceleration
Taylor series (ek point par saari higher derivatives use karta hai)
Trigonometric derivatives

Order	Lagrange	Leibniz	Newton (time)
1st	$f^{'} (x)$	$\frac{d y}{d x}$	$\overset{y}{˙}$
2nd	$f^{''} (x)$	$\frac{d ^{2} y}{d x ^{2}}$	$\overset{y}{¨}$
3rd	$f^{'''} (x)$	$\frac{d ^{3} y}{d x ^{3}}$	$y ...$
$n$ -th	$f^{(n)} (x)$	$\frac{d ^{n} y}{d x ^{n}}$	—