Hosila va differensiallash qoidalari

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

Hosila tushunchasi

• Egri chiziqning nuqtadagi gradienti = shu nuqtadagi urinmaning gradienti. Uni nuqtaga tobora yaqinlashuvchi vatarlar gradientlari orqali topish mumkin.
• Birinchi tamoyillardan (first principles): $\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}$
• Belgilash: birinchi hosila $\dfrac{dy}{dx}$ yoki $f'(x)$; ikkinchi hosila $\dfrac{d^2y}{dx^2}$ yoki $f''(x)$.

Standart natijalar ($k$, $n$, $c$ — o'zgarmaslar)

• ⭐ $y = kx^n \;\Rightarrow\; \dfrac{dy}{dx} = knx^{n-1}$ ($n$ — istalgan son: manfiy yoki kasr ham)
• $y = c \;\Rightarrow\; \dfrac{dy}{dx} = 0$
• ⭐ Yig'indi: $y = f(x) + g(x) \;\Rightarrow\; \dfrac{dy}{dx} = f'(x) + g'(x)$
• ⭐ Differensiallashdan oldin ildiz va kasrlarni daraja shakliga keltiring: $\sqrt{x} = x^{\frac{1}{2}} \qquad \frac{1}{x} = x^{-1} \qquad \frac{k}{x^n} = kx^{-n} \qquad \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$
• Ko'paytma yoki bo'linma bo'lsa, avval qavslarni oching yoki bo'lib chiqing: $(x + 1)(x - 5) = x^2 - 4x - 5$.

🟢 Boshqa shakl

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Misol: $f(x) = x^2$ uchun $\dfrac{(x + h)^2 - x^2}{h} = 2x + h \to 2x$.

🟢 Teskari hosila

$\frac{dy}{dx} = \frac{1}{\;\dfrac{dx}{dy}\;}$

🔵 Yuqori tartibli hosilalar

• $x^n$ ning $n$-hosilasi: $n!$; $(n+1)$-hosilasi: $0$
• $n$-darajali ko'phadning $(n + 1)$-hosilasi nolga teng.

🟡 Qoidalar

$\text{Ko'paytma: } (uv)' = u'v + uv' \qquad \text{Bo'linma: } \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

🟡 Standart hosilalar