Derivatives of all six trig functions

4.1.18 · Maths › Calculus I — Limits & Derivatives

Hum care kyun karte hain?

KYUN: Trig functions kisi bhi cheez ko model karte hain jo oscillate karti hai — springs, waves, AC current, circular motion. Derivative tumhe us oscillation ki rate batata hai (ek jhoolte pendulum ki velocity, ek wave ki slope). Agar tum $\sin$ aur $\cos$ differentiate nahi kar sakte, toh tum physics, signals, ya differential equations nahi kar sakte.

KYA: Ek aisa rule jo chhaon trig functions mein se kisi ko bhi leta hai aur uski instantaneous slope return karta hai.

KAISE: ${\sin }^{\prime }$ aur ${\cos }^{\prime }$ ko limit definition + do special limits se banao, phir quotient rule se baaki chaar derive karo. Koi bhi aisi table mat yaad karo jo tum rebuild nahi kar sakte.

Step 0 — Do limits jo hume pehle establish karne hain

Pehla limit sach kyun hai (geometric squeeze): Unit circle mein ek chote angle $h$ ke liye (radians mein), arc length $h$ hai, chord/sine $\sin h$ hai, aur tangent $\tan h$ hai. Geometry se milta hai: $\sin h\le h\le \tan h.$ $\sin h$ se divide karo aur flip karo: $\cos h\le \frac{\sin h}{h}\le 1.$ Jaise $h\to 0$, $\cos h\to 1$, toh squeeze force karta hai $\frac{\sin h}{h}\to 1$.

Doosra limit sach kyun hai (algebra trick): Conjugate se multiply karo: $\frac{\cos h-1}{h}\cdot \frac{\cos h+1}{\cos h+1}=\frac{{\cos }^{2}h-1}{h(\cos h+1)}=\frac{-{\sin }^{2}h}{h(\cos h+1)}=-\frac{\sin h}{h}\cdot \frac{\sin h}{\cos h+1}.$ Jaise $h\to 0$: $\frac{\sin h}{h}\to 1$ aur $\frac{\sin h}{\cos h+1}\to \frac{0}{2}=0$. Product $\to 1\cdot 0=0$. ✅

Step 1 — Limit definition se $\frac{d}{dx}\sin x$ derive karo

$\frac{d}{dx}\sin x={\lim }_{h\to 0}\frac{\sin (x+h)-\sin x}{h}$

Ye step kyun? Ye derivative ki definition hai; abhi tak kuch trig nahi.

Angle-addition identity $\sin (x+h)=\sin x\cos h+\cos x\sin h$ use karo: $={\lim }_{h\to 0}\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}$ $={\lim }_{h\to 0}\left[\sin x\cdot \underset{\to 0}{\underbrace{\frac{\cos h-1}{h}}}+\cos x\cdot \underset{\to 1}{\underbrace{\frac{\sin h}{h}}}\right]$

Ye step kyun? $\sin x$ aur $\cos x$ terms ko group karo taaki do foundation limits appear hon.

$=\sin x\cdot 0+\cos x\cdot 1=\boxed{\cos x}$

Step 2 — Same engine se $\frac{d}{dx}\cos x$ milta hai

$\cos (x+h)=\cos x\cos h-\sin x\sin h$ use karke: $\frac{d}{dx}\cos x={\lim }_{h\to 0}\left[\cos x\cdot \frac{\cos h-1}{h}-\sin x\cdot \frac{\sin h}{h}\right]=\cos x\cdot 0-\sin x\cdot 1=\boxed{-\sin x}$

Minus sign yaad rakhne wali ek cheez hai: $\cos$ differentiate karne ki ek "cost" hai — negative.

Step 3 — Quotient rule se baaki chaar derive karo

Quotient rule yaad karo: ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$.

$\tan x=\frac{\sin x}{\cos x}$

$\frac{d}{dx}\tan x=\frac{\cos x\cdot \cos x-\sin x\cdot (-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}=\boxed{{\sec }^{2}x}$ Ye step kyun? Numerator Pythagorean identity ${\sin }^2+{\cos }^2=1$ se collapse ho jaata hai.

$\cot x=\frac{\cos x}{\sin x}$

$\frac{d}{dx}\cot x=\frac{-\sin x\cdot \sin x-\cos x\cdot \cos x}{{\sin }^{2}x}=\frac{-({\sin }^{2}x+{\cos }^{2}x)}{{\sin }^{2}x}=\boxed{-{\csc }^{2}x}$

$\sec x=\frac{1}{\cos x}$

$\frac{d}{dx}\sec x=\frac{0\cdot \cos x-1\cdot (-\sin x)}{{\cos }^{2}x}=\frac{\sin x}{{\cos }^{2}x}=\frac{1}{\cos x}\cdot \frac{\sin x}{\cos x}=\boxed{\sec x\tan x}$

$\csc x=\frac{1}{\sin x}$

$\frac{d}{dx}\csc x=\frac{0\cdot \sin x-1\cdot \cos x}{{\sin }^{2}x}=\frac{-\cos x}{{\sin }^{2}x}=\boxed{-\csc x\cot x}$

Worked examples

Flashcards

Connections

• 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 (physics application)
• Derivatives of inverse trig functions

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps