Parametric differentiation — dy - dx, d²y - dx²

4.1.23 · Maths › Calculus I — Limits & Derivatives

WHY — slope bina $y=f(x)$ ke

HOW — first derivative (scratch se derive kiya)

Hum chain rule se shuru karte hain, yahi ek cheez hame chahiye.

$y$ depend karta hai $x$ pe, aur $x$ depend karta hai $t$ pe: $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$

Yeh sirf chain rule hai jo $y$ pe apply kiya gaya hai $t$ ke function ke roop mein through $x$. Ab jo hum chahte hain uske liye solve karo, dono sides ko $\frac{dx}{dt}$ se divide karo (allowed jab $\frac{dx}{dt}\ne 0$ ho):

HOW — second derivative (woh part jahan sabse zyada log galti karte hain)

Derivation. Maano $y^{\prime }\equiv \frac{dy}{dx}$ ($t$ ka function hai). Definition se: $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}(y^{\prime })$

Lekin hum sirf $t$ ke w.r.t. cheezein differentiate karna jaante hain. $y^{\prime }$ pe wohi parametric trick use karo: $\frac{d}{dx}(y^{\prime })=\frac{d(y^{\prime })/dt}{dx/dt}$

Ab $y^{\prime }=\dot{y}/\dot{x}$ hai, toh ise $t$ ke w.r.t. quotient rule se differentiate karo: $\frac{d}{dt}\left(\frac{\dot{y}}{\dot{x}}\right)=\frac{\ddot{y}\dot{x}-\dot{y}\ddot{x}}{{\dot{x}}^2}$

$\dot{x}$ se divide karo:

Worked examples

Active recall

Connections

• Chain rule — dono formulas ke peeche ka engine.
• Quotient rule — $\dot{y}/\dot{x}$ ko differentiate karne ke liye use hota hai.
• Implicit differentiation — alternative jab koi parameter available na ho.
• Tangents and normals — slope $dy/dx$ parametric curves pe tangent lines deta hai.
• Concavity and second derivative — $d^{2}y/d{x}^2$ ka sign concave up/down batata hai.
• Cycloid, Parametric curves — is technique ke natural ghar.

Concept Map