Zanjir qoidasi va oʻzgarish tezliklari

⭐ — imtihonda eng ko'p ishlatiladigan formulalar. · 🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Zanjir qoidasi (chain rule)

• ⭐ $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

Bog'langan o'zgarish tezliklari (connected rates of change)

• ⭐ $\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$

Misol: doira radiusi $\dfrac{dr}{dt} = 0.5$ tezlik bilan o'smoqda. $A = \pi r^2$, $r = 4$ da: $\dfrac{dA}{dt} = 2\pi r \times 0.5 = 4\pi$.

Zanjir qoidasining tayyor shakli

$\frac{d}{dx}(ax + b)^n = an(ax + b)^{n-1} \qquad \frac{d}{dx}\big[f(x)\big]^n = n\big[f(x)\big]^{n-1} f'(x)$

Misol: $y = (3x - 1)^4 \;\Rightarrow\; \dfrac{dy}{dx} = 4 \cdot 3\,(3x - 1)^3 = 12(3x - 1)^3$

🟢 Taqribiy hisoblash

$\delta y \approx \frac{dy}{dx}\,\delta x$

🔵 Foiz xatolik

$y = kx^n$ bo'lsa:

$\frac{\delta y}{y} \approx n\,\frac{\delta x}{x}$

Misol: shar radiusi $1\%$ ga oshsa, hajmi ($V \propto r^3$) taxminan $3\%$ ga oshadi.

🟢 Kinematika (mexanika bilan bog'liq)

$v = \frac{ds}{dt} \qquad a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

• Jism to'xtaydi: $v = 0$; eng katta tezlik: $a = 0$.

🟡 Zanjir qoidasining umumiy shakllari

$\frac{d}{dx}e^{f(x)} = f'(x)\,e^{f(x)} \qquad \frac{d}{dx}\ln f(x) = \frac{f'(x)}{f(x)} \qquad \frac{d}{dx}\sin f(x) = f'(x)\cos f(x)$

🟡 Oshkormas va parametrik differensiallash

• Oshkormas: $\dfrac{d}{dx}(y^2) = 2y\,\dfrac{dy}{dx}$, $\quad \dfrac{d}{dx}(xy) = y + x\,\dfrac{dy}{dx}$
• Parametrik ($x = x(t)$, $y = y(t)$): $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$