Foundations — Parametric differentiation — dy - dx, d²y - dx²

4.1.23 · D1 · Maths › Calculus I — Limits & Derivatives › Parametric differentiation — dy - dx, d²y - dx²

Yeh page ground floor hai. Parent topic freely use karta hai symbols jaise $t$, $\dot{x}$, $\frac{dy}{dx}$, $\frac{d^{2}y}{d{x}^2}$ aur words jaise slope, quotient rule, chain rule. Neeche, har ek ko zero se build kiya gaya hai, ek aisi order mein jahan har item sirf pehle walon pe lean karta hai. Agar tum neeche di gayi checklist pass kar sako, toh parent note ek aisi kahani ki tarah padhega jo tumhe pehle se pata hai.

0. Point $(x,y)$ — "plane par ek jagah" kahan se aata hai

Figure dekho: horizontal line $x$-axis hai (rightward = bada $x$), vertical line $y$-axis hai (upward = bada $y$). Unka crossing point origin $(0,0)$ hai. Ek point dhundha jaata hai $x$ steps across chaltey hue, phir $y$ steps upar.

Topic ko yeh kyun chahiye: pura game ek curve describe karna hai, aur curve bas points $(x,y)$ ka ek crowd hai. Baaki sab is baare mein hai ki woh points kaise choose hote hain aur unke beech ka trail kitna steep hai.

1. Ek function $x(t)$ — ek machine jo ek number ko doosre mein convert karti hai

Ek vending machine socho: button $t=2$ dabaao, ek specific snack nikalti hai. $t=2$ phir dabaao, tumhe wahi snack milti hai — woh reliability hi ise function banati hai.

Topic ko yeh kyun chahiye: parametric form mein hamare paas do machines hain, $x(t)$ aur $y(t)$, same input $t$ se fed. Woh dono milke decide karte hain dot kahan hai.

2. Parameter $t$ — hidden dial (socho: time)

Figure mein, jaise $t$ increase hota hai orange dot circle ke around march karta hai. $t=0$ par woh right mein baith ta hai; quarter-turn baad woh top par hai. Same $x=0$ do baar hota hai (top aur bottom) — proof ki plain $y=f(x)$ ise describe nahi kar sakta, lekin $t$ kar sakta hai.

Topic ko yeh kyun chahiye: $t$ ke bina "$x$ ki speed" ya "$y$ ki speed" compare karne ka koi zariya nahi — aur woh comparison hi slope hai.

3. Slope — "rise over run", woh number jo steepness matlab rakhta hai

Figure mein line par sawaar ek right triangle dikhta hai: horizontal leg run hai, vertical leg rise hai. Unka ratio slope hai, aur woh same rehta hai chahe triangle kitna bhi bada draw karo — woh constancy hi "slope" ko ek meaningful number banati hai.

Topic ko yeh kyun chahiye: $\frac{dy}{dx}$ hi slope hai. Poora first derivative banaya gaya hai is sawaal ka jawab dene ke liye ki "curve yahan kitna steep hai?"

4. Symbol $\Delta$ aur $d$ ki taraf shrink — ek chunk se ek instant tak

Curve mein microscope se zoom karne ki imagine karo. Ek finite $\Delta$-triangle ek bend ko straddle karta hai aur steepness ko sirf approximate karta hai. Use $d$-triangle tak shrink karo aur woh curve ko itna tightly hug karta hai ki ek single point par steepness capture karta hai. Yeh point-tak-shrinking limit ka kaam hai — wajah ki yeh chapter Limits & Derivatives kehlaata hai.

Topic ko yeh kyun chahiye: $\frac{dy}{dx}$ us infinitesimal triangle ka slope hai. Parent mein line "$dx=\dot{x}dt$" bas yeh differential idea hai dial $t$ ke through apply hua.

5. Derivative $\frac{dx}{dt}$ aur dot $\dot{x}$ — ek machine ki speed

Topic ko yeh kyun chahiye: first derivative hi $\dot{y}/\dot{x}$ hai — vertical speed divided by horizontal speed. Sab kuch inhi do dotted quantities par hinge karta hai.

6. Chain rule — ek middle-man ke through differentiate kaise karein

Topic ko yeh kyun chahiye: parent derive karta hai slope formula $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$ likh ke aur $\frac{dy}{dx}$ solve karke. Chain rule nahi, formula nahi. Poori details Chain rule mein hain.

7. Quotient rule — ek fraction ko differentiate karna

Topic ko yeh kyun chahiye: second derivative fraction $\frac{\dot{y}}{\dot{x}}$ ko differentiate karta hai. Parent exactly yeh rule apply karta hai $u=\dot{y}$, $v=\dot{x}$ ke saath, upar $\ddot{y}\dot{x}-\dot{y}\ddot{x}$ produce karta hai aur neeche ${\dot{x}}^2$ — $\dot{x}$ se ek aur divide karne se famous ${\dot{x}}^3$ milta hai. Zyada Quotient rule par.

8. Second derivative $\frac{d^{2}y}{d{x}^2}$ — slope khud kaise bend karta hai

Figure mein, smile ki tarah bend karne wala curve (paani rakhne wala) increasing slope rakhta hai — woh positive $\frac{d^{2}y}{d{x}^2}$ hai, jise concave up kehte hain. Frown shape ka slope decreasing hai — negative, concave down. Sign hi poori baat hai, Concavity and second derivative mein explore kiya gaya.

Topic ko yeh kyun chahiye: yeh second headline formula hai. Iska sahi derivation demand karta hai chain rule aur quotient rule aur differential idea — upar ke har prerequisite, stacked.

Foundations topic ko kaise feed karte hain

Equipment checklist

Right side cover karo; reveal karne se pehle jawab zor se bolo.

Connections

• Chain rule — dono formulas ke peeche rates-multiply engine.
• Quotient rule — second derivative ke liye fraction $\dot{y}/\dot{x}$ differentiate karta hai.
• Implicit differentiation — ek cousin technique jab koi parameter nahi hota.
• Tangents and normals — woh slope use karta hai jo tum yahan build karte ho.
• Concavity and second derivative — $\frac{d^{2}y}{d{x}^2}$ ka sign padhta hai.
• Parametric curves, Cycloid — jahan yeh symbols life mein aate hain.