Foundations — Derivatives of inverse trig functions — all six

4.1.21 · D1 · Maths › Calculus I — Limits & Derivatives › Derivatives of inverse trig functions — all six

Pehle aap parent page par jo chhe inverse-trig derivatives hain unhe derive kar sako, uske liye aapko har woh symbol apna banana hoga jo woh aap par phenkta hai. Yeh page har ek ko scratch se banata hai — seedhe words, phir ek picture, phir yeh topic isko kyun chahta hai. Upar se neeche padho; har block uske upar wale par tikaa hai.

1. Angle — raw input

Topic ko yeh kyun chahiye. $\arcsin x$ literally ek angle output karta hai. Agar tumhare dimag mein "angle = turning ki maatra" ki picture nahi hai, toh "woh angle jiska sine $x$ hai" wali baat sirf shor hai.

Hum turning ko radians mein measure karte hain — ek unit jisme poora aadha turn $\pi \approx 3.14$ hota hai (na ki $180$). Radians kyun, degrees kyun nahi? Kyunki calculus ke formulas jaise $\frac{d}{dx}\sin x=\cos x$ sirf radians mein saaf hote hain. Yeh promise dekho Derivatives of Trig Functions mein.

2. Right triangle — jahan ratios rehte hain

Topic ko yeh kyun chahiye. Har inverse-trig derivation ke end mein theek yehi triangle draw hoti hai taaki kisi mushkil $\cos y$ ya $\sec y$ ko $x$ wali expression mein badla ja sake. Triangle ek translator hai.

3. sin, cos, tan — ratio machines

Topic ko yeh kyun chahiye. Parent jo equation se shuru karta hai — $\sin y=x$ — woh kuch nahi balki "opposite-over-hypotenuse ratio $x$ ke barabar hai." Aage jo bhi hai woh usi sentence se slope nikaalna hai.

Baaki teen ratios bas reciprocals hain: $\csc y=\frac{1}{\sin y},\sec y=\frac{1}{\cos y},\cot y=\frac{1}{\tan y}.$

4. Inverse function — machine ko ulta chalana

Topic ko yeh kyun chahiye. Poora chapter inhi ulte machines ko differentiate karne ke baare mein hai. Bridge line $y=\arcsin x⟺\sin y=x$ woh hai jo humein aasaan side ko differentiate karne deti hai. Poori story ke liye dekho Inverse Functions and their Domains.

5. Domain aur range restriction — kyun signs aisa behave karti hain

$\sin$ baar baar repeat hota hai, toh infinitely many angles ek hi sine share karte hain. $\arcsin$ ko ek genuine ek-jawab wali machine banane ke liye, hum uski output range ko $[-\frac{\pi }{2},\frac{\pi }{2}]$ par restrict karte hain. Us window ke andar ek baat sone ke barabar hai: $\cos y\ge 0$.

Topic ko yeh kyun chahiye. Yahi "$\cos y\ge 0$" wajah hai ki parent positive square root leta hai: $\cos y=+\sqrt{1-x^2}$. Range badlo (jaise arccos karta hai, $[0,\pi ]$ use karke) aur ek sign palat jaata hai. Chhhon formulas mein saari signs yahan se decide hoti hain — dekho Inverse Functions and their Domains.

6. Pythagorean identity — side-length ka kanoon

Topic ko yeh kyun chahiye. Solve karne ke baad, parent ko milta hai $\frac{dy}{dx}=\frac{1}{\cos y}$ — $y$ mein jawab, lekin hum chahte hain $x$ mein jawab. Identity $\cos y=\sqrt{1-{\sin }^{2}y}=\sqrt{1-x^2}$ likh deti hai. Saaf $\arctan$ result poori tarah $1+{\tan }^{2}y={\sec }^{2}y$ par tika hai. Poora toolkit: Pythagorean Identities.

7. Derivative aur $\frac{dy}{dx}$ notation — "output kitni tezi se badalta hai"

Topic ko yeh kyun chahiye. Poora page poochh raha hai "ratio badalne par angle wala jawab kitni tezi se badalta hai?" — woh hai $\frac{d}{dx}\arcsin x$. Derivative nahi toh topic nahi.

8. Chain rule — machine-ke-andar-machine ko differentiate karna

Topic ko yeh kyun chahiye. $\sin y=x$ ko differentiate karna jahan $y$ secretly $x$ par depend karta hai, milta hai $\cos y\cdot \frac{dy}{dx}=1$ — $\frac{dy}{dx}$ aata hi chain rule ki wajah se hai. Yahi woh source bhi hai jahan se $\frac{d}{dx}\arctan (3x)=\frac{3}{1+9x^2}$ mein "extra $3$" aata hai. Deep dive: Chain Rule.

9. Implicit differentiation — poori equation differentiate karo, phir solve karo

Topic ko yeh kyun chahiye. Yahi engine hai. Hum $\arcsin$ ko seedha differentiate nahi kar sakte, lekin hum $\sin y=x$ ko differentiate karke solve kar sakte hain. Chhhon formulas mein se har ek yahi ek trick baar baar use karta hai. Poora method: Implicit Differentiation.

Prerequisite map

Equipment checklist

Connections

• Implicit Differentiation — woh engine jo $\sin y=x$ ko slope mein badal deta hai
• Chain Rule — $\frac{dy}{dx}$ kyun aata hai aur "extra factor" kahan se aata hai
• Pythagorean Identities — $\cos y,\sec y$ ko wapas $x$ mein convert karta hai
• Inverse Functions and their Domains — woh range limits jo har sign fix karti hain
• Derivatives of Trig Functions — forward derivatives jo hum ulte chalate hain
• Integration by Inverse Trig — jahan yeh formulas ulte use hote hain