Derivatives of all six trig functions: Question bank

4.1.18 · D5 · Maths › Calculus I — Limits & Derivatives › Derivatives of all six trig functions

Shuru karne se pehle, neeche baar baar aane wale teen words ko pin down karte hain taaki kuch bhi bina samjhe use na ho:

Sach ya jhooth — justify karo

T/F jawab do, phir reveal karne se pehle apne words mein reason do.

\frac{d}{d x} sin x = cos x

chahe

x

degrees mein ho ya radians mein — dono mein kaam karta hai.

Jhooth (False). Yeh sirf radians mein kaam karta hai. Degrees mein

sin h \approx \frac{π}{180} h

, toh derivative mein

\frac{π}{180}

ka factor aa jaata hai:

\frac{d}{d x} sin x^{\circ} = \frac{π}{180} cos x^{\circ}

Kyunki

\frac{d}{d x} sin x = cos x

aur

\frac{d}{d x} cos x = - sin x

, yeh pair "rotate" karta hai —

\frac{d}{d x}

ko chaar baar lagaane par wapas

sin x

mil jaata hai.

Sach (True).

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

. Chaar derivatives ek poora cycle hai, aur isliye

sin

aur

cos

Simple Harmonic Motion mein oscillator equation solve karte hain.

\frac{d}{d x} tan x = sec^{2} x

saare real

x

ke liye valid hai.

Jhooth (False).

tan x

aur

sec^{2} x

dono wahan blow up karte hain jahan

cos x = 0

ho (yaani

x = \frac{π}{2} + k π

par). Rule sirf usi domain par kaam karta hai jahan

tan x

exist karta ho.

Pythagorean identity

tan^{'}

derive karne mein use hoti hai lekin

sin^{'}

derive karne mein nahi.

Sach (True).

tan^{'}

ko numerator collapse karne ke liye

cos^{2} + sin^{2} = 1

chahiye.

sin^{'}

ki derivation mein do foundation limits aur Angle addition formulas use hote hain — koi Pythagoras nahi.

sec^{2} x \geq 1

domain mein har

x

ke liye hota hai, toh

tan x

hamesha increase karta hai.

Sach (True).

sec^{2} x = \frac{1}{c o s ^{2} x} \geq 1

kyunki

cos^{2} x \leq 1

. Ek slope jo kabhi

1

se neeche nahi jaata iska matlab hai

tan

apne vertical asymptotes ke beech har interval par chadhta hi rehta hai.

Kyunki

cos

ek even function hai (

cos (- x) = cos x

), toh iska derivative

- sin x

bhi even hai.

Jhooth (False). Ek even function ka derivative odd hota hai.

- sin (- x) = sin x = - (- sin x)

, toh

- sin

odd hai — yeh us rule se match karta hai ki differentiate karne par even

\leftrightarrow

odd parity flip ho jaati hai.

\frac{d}{d x} csc x = - csc x cot x

x = \frac{π}{2}

par check kar sakte hain: iska result

0

hona chahiye.

Sach (True).

x = \frac{π}{2}

par,

cot (\frac{π}{2}) = \frac{c o s}{s i n} = \frac{0}{1} = 0

, toh derivative

0

hai. Geometrically

csc

wahan minimum par hai (value

1

), aur minimum ka slope zero hota hai.

Error dhundo

Har line mein ek galat step hai. Use dhundo, phir fix reveal karo.

\frac{d}{d x} sin (x^{2}) = cos (x^{2})

Inside derivative missing hai (Chain Rule). Inside

x^{2}

ko bhi differentiate karna hota hai: sahi jawab hai

cos (x^{2}) \cdot 2 x = 2 x cos (x^{2})

\frac{d}{d x} cos (5 x) = - sin (5 x)

Inner function ka factor

5

bhool gaye. Sahi:

- 5 sin (5 x)

. Table rule

cos^{'} = - sin

sirf tab apply hota hai jab andar plain

x

ho.

\frac{d}{d x} sec x = - sec x tan x

Sign galat hai.

sec

mein koi "co" nahi, toh yeh positive hai:

\frac{d}{d x} sec x = + sec x tan x

. Minus

csc

ke saath jaata hai.

\frac{d}{d x} tan x = sec x^{2}

Notation abuse hai:

sec x^{2}

sec (x^{2})

padha jaata hai. Jawab

sec^{2} x

hai, yaani

(sec x)^{2}

. Agar confusion ho toh

(sec x)^{2}

likh lo.

\frac{d}{d x} (sin x cos x) = cos x \cdot (- sin x)

Ise chain rule ki tarah treat kiya; actually yeh ek product hai (Product Rule):

cos x cos x + sin x (- sin x) = cos^{2} x - sin^{2} x

\frac{d}{d x} cot x

ke liye:

\frac{c o s x}{s i n x}

par quotient rule lagane se milta hai

\frac{- s i n x s i n x + c o s x c o s x}{s i n ^{2} x} = csc^{2} x

Numerator mein sign ki galti hai. Quotient Rule deta hai

\frac{( - s i n x ) ( s i n x ) - ( c o s x ) ( c o s x )}{s i n ^{2} x} = \frac{- ( s i n ^{2} + c o s ^{2} )}{s i n ^{2} x} = - csc^{2} x

\frac{d}{d x} sin x = cos x

, toh

\frac{d}{d x} sin (3 0^{\circ}) = cos (3 0^{\circ})

Do errors hain:

sin (3 0^{\circ})

ek constant hai, toh iska derivative

0

hai; aur agar input variable-degree bhi hota toh

\frac{π}{180}

factor ki zaroorat padti. Kisi bhi constant ka derivative

0

hota hai.

"Kyun" waale sawaal

Kyun angle radians mein hona chahiye taaki

\frac{d}{d x} sin x = cos x

ho?

Poori derivation

lim_{h \to 0} \frac{s i n h}{h} = 1

par tikhi hai, aur woh limit

1

sirf tab hoti hai jab arc length angle measure ke barabar ho — jo radians ki definition hi hai. Degrees angle ko rescale kar dete hain, jisse

\frac{π}{180}

aa jaata hai.

Kyun exactly teen derivatives mein minus sign hota hai, chhon mein se?

Teen "co-" functions (

cos, cot, csc

) apne partners ke badhne ke saath

0

ke paas decrease karte hain. Structurally, har "co" derivative numerator mein

cos

ko differentiate karne se aata hai (jisse

- sin

milta hai) ya

sin

ke reciprocal se, aur woh

cos^{'} = - sin

wala minus propagate ho jaata hai.

Kyun hum

tan, cot, sec, csc

derive kar sakte hain bina limit definition dobara use kiye?

Har ek

sin

aur

cos

ka ratio hai, jinke derivatives hum pehle se earn kar chuke hain. Quotient Rule "ratio ki slope" ko known slopes mein convert kar deta hai — koi nayi limits nahi chahiye.

Kyun

\frac{d}{d x} tan x = sec^{2} x

kabhi negative value nahi leta?

sec^{2} x = 1/ cos^{2} x

ek square divided by

1

hai, toh yeh hamesha positive hota hai. Yeh encode karta hai ki

tan

kabhi bhi sirf chadh'ta hi hai — asymptotes ke beech kabhi nahi muda.

Kyun

cos x

sin x

ki rate hai, sirf ek "related wave" nahi?

Kyunki

cos x

literally

\frac{s i n ( x + h ) - s i n x}{h}

ka limit hai —

sin

mein instantaneous change. Dekho Limit definition of the derivative. Phase shift (

cos

sin

\frac{π}{2}

aage hai) is rate ka visual representation hai.

Kyun inverse trig derivatives (jaise

\frac{d}{d x} arctan x = \frac{1}{1 + x ^{2}}

) algebraic lagte hain, bina kisi trig ke?

Relationship ko invert karne par

sec^{2}

aur doston ko Pythagorean identity sin^2 + cos^2 = 1 ke zariye

x

ke terms mein rewrite karna padta hai, jisse trig dissolve ho jaata hai. Details Derivatives of inverse trig functions mein hain.

Edge cases

x = 0

par

sin x

ki slope kya hai, aur woh dikhti kaisi hai?

cos (0) = 1

—

sin

wave jitni steep kabhi chadh sakti hai. Origin ke paas

sin x \approx x

, jo slope

1

ki ek line hai — yeh exactly woh foundation limit hai jo visible ho rahi hai.

x \to \frac{π}{2}^{-}

\frac{d}{d x} tan x

ka kya hoga?

sec^{2} x \to + \infty

. Tangent curve apne asymptote ke paas vertical ho jaata hai; infinite slope graph ke upar shoot karne se match karta hai.

Kya

\frac{d}{d x} cot x

x = 0

par defined hai?

Nahi.

cot x = cos x / sin x

wahan undefined hai jahan

sin x = 0

ho (including

x = 0

), aur

- csc^{2} x = - 1/ sin^{2} x

wahan blow up karta hai. Derivative parent ke domain gaps share karta hai.

x = 0

par

\frac{d}{d x} sec x

— value aur meaning?

sec (0) tan (0) = 1 \cdot 0 = 0

sec

x = 0

par local minimum

1

hai, aur minimum ki slope zero hoti hai — consistent hai.

Kya

\frac{d}{d x} sin x = cos x

peaks

x = \frac{π}{2}

par bhi hold karta hai jahan

sin

flat hoti hai?

Haan.

cos (\frac{π}{2}) = 0

, crest par zero slope sahi report karta hai. Rule maxima/minima ko automatically handle karta hai — koi special case nahi chahiye.

x \to 0^{+}

\frac{d}{d x} csc x

ka kya hoga?

- csc x cot x = - \frac{c o s x}{s i n ^{2} x} \to - \infty

. Jaise

csc x

0^{+}

ke paas

+ \infty

ki taraf rocket karta hai, uski slope theek pehle steeply negative hoti hai — asymptote mein ek bada downward drop right side se.

Agar koi function constant hai jaise

y = tan (π /4) = 1

, toh uska derivative kya hai?

0

. Jab input ek fixed number ho, output constant hoti hai, aur constants ki slope zero hoti hai — koi bhi trig rule fire nahi karta.

Recall Har trap ki ek-line summary

Sirf radians · chain rule ko andar ki cheez chahiye · "co" ⇒ minus · derivative rules parent ke domain gaps inherit karte hain · constants ko differentiate karne par $0$ milta hai.

Connections

Derivatives of all six trig functions (parent)
Limit definition of the derivative
The Squeeze Theorem
Quotient Rule
Chain Rule
Product Rule
Pythagorean identity sin^2 + cos^2 = 1
Angle addition formulas
Simple Harmonic Motion
Derivatives of inverse trig functions