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Hosila - funktsiya qiymatining o'zgarishining mustaqil o'zgaruvchidagi o'zgarishiga nisbati.
Funksiyaning qaysidir nuqtadagi hosilalari funksiyaning shu nuqtadagi o‘zgarish tezligini bildiradi. O'zgarish tezligini \(\Delta y\) funktsiyasining mustaqil o'zgaruvchining \(\Delta x\) o'zgarishiga o'zgarish tezligi bilan hisoblash mumkin, bu nisbat chegarada \(\Delta x \to 0\) sifatida qabul qilinadi. \(\Delta x \to 0\) . f(x) funksiyaning hosilasi uning o‘zgarish tezligini ifodalaydi va \(f\prime(x) \) yoki df ∕ dx bilan belgilanadi.

Keling, avval uning ta'rifini va hosilaning rasmli tasvirini ko'rib chiqaylik.

f ning hosilasi - f ning o'zgarish tezligi. Yuqoridagi egri chiziqning grafigiga qarang. U f(x) ning ikkita x va \(x + \Delta x \) nuqtalarida mos ravishda f(x) va \(f(x + \Delta x)\) qiymatini ifodalaydi. Ushbu ikki nuqta orasidagi intervalni cheksiz kichik bo'lguncha kichraytirsangiz, bizda \(\Delta x \to 0\) chegarasi mavjud.


\(f\prime(x) = \frac{df }{dx} = \lim\limits_{ \Delta x\to0}\frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim\limits_{ \Delta x\to0}\frac{\Delta f }{\Delta x}\)

Numerator \(f(x + \Delta x) - f(x)\) f funktsiya qiymatining \(\Delta x\) oralig'ida tegishli o'zgarishini ifodalaydi. Bu f funktsiyaning x nuqtadagi hosilasini, f ning shu nuqtadagi o'zgarish tezligini hosil qiladi.

f(x) funksiyaning x nuqtada hosilasini topish bosqichlari:

1. Farq nisbatini hosil qiling \(\frac{dy }{dx} = \frac{f(x+\Delta x) - f(x)}{\Delta x}\)
2. Iloji boricha bekor qilib, qismni soddalashtiring.
3. Ko'rsatkichga chegara qo'llagan holda \(f\prime(x)\) hosilasini toping. Agar bu chegara mavjud bo'lsa, f(x) funksiya x da differentsiallanadi deymiz.

Keling, bir nechta funktsiyalar uchun hosilalarni olishga harakat qilaylik

1-misol : y = x funksiyaning hosilasini hisoblang

\(y\prime(x) = \lim\limits_{ \Delta x\to0}\frac{\Delta y}{\Delta x} = \lim\limits_{\Delta x\to0} \frac{(x+\Delta x) - x}{\Delta x} = \lim\limits_{ \Delta x\to0}\frac{\Delta x}{\Delta x} = \lim\limits_{ \Delta x\to0}1=1\)


2-misol: f(x) = 5x + 2 funksiyaning hosilasini toping

bu 5x + 2 funksiyasining chizmasi

\(\Delta y = y(x+\Delta x) - y(x) = [5(x+\Delta x) +2] - [5x +2] = 5x + 5\Delta x + 2 - 5x - 2 = 5\Delta x\)

Farq nisbati \(\frac{5\Delta x}{\Delta x}\) = 5

\(\therefore\textrm{ Hosil } f\prime(x) = \lim\limits_{ \Delta x\to0}5 = 5\)

3-misol: f(x) = x 2 kvadrat tenglamaning hosilasini toping. Keling, grafikdan foydalanamiz va hosilalarni yaxshiroq tushunamiz.

f(x) = x 2

\(\Delta y = y(x+\Delta x) - y(x) = (x+\Delta x)^2 - x^2\)

\(\Delta y = x^2 + 2x\Delta x +\Delta x^2-x^2 = \Delta x(2x+\Delta x)\)

X 2 ning hosilasi 2x ga teng. Bu shuni anglatadiki, x 2 funktsiyasi uchun har qanday nuqtadagi o'zgarish tezligi 2x ga teng.

f ning x = 2 da o'zgarish tezligi, x = 2 da \(f\prime(x)\) qiymati, ya'ni \(f\prime (x) = 4\)

Umumiy funksiyalarning hosilalari

Umumiy funksiya funksiya Hosil
konstanta c 0
Chiziq x 1
  ax a
Kvadrat x 2 2x
Kvadrat ildiz \(\sqrt x\) \(\frac{1}{2}x^{-1/2}\)
Eksponensial e x e x
Logarifmlar \(\log_a(x)\) 1/(x In(a))
Trigonometriya(x radianda) \(\sin(x)\) \(\cos(x) \)
  \(\cos(x)\) \(-\sin(x)\)
  \(\tan(x)\)

\(\sec^2(x)\)

Bu yerda ko‘p funksiyalarning hosilalarini ishlab chiqishda yordam beradigan foydali qoidalar mavjud:

  • Doimiy qoida: f(x) = c keyin \(f\prime(x) = 0\)
  • Doimiy ko‘p qoida: \(g(x) = c \cdot f(x) \textrm{ keyin } g\prime(x) = c \cdot f\prime(x)\)
  • Quvvat qoidasi: \( f(x) = x^n \textrm{ keyin } f\prime(x) = nx^{n-1}\)
  • Yig‘indi va farq qoidasi: \(h(x) = f(x) \pm g(x) \textrm{ keyin } h\prime(x) = f\prime(x) \pm g\prime(x)\)
  • Mahsulot qoidasi: \(h(x) = f(x)g(x) \textrm{ keyin } h\prime(x) = f\prime(x)g(x) + f(x)g\prime(x)\)
  • Qism qoidasi: \(h(x) = \frac{f(x)}{g(x)} \textrm{ keyin } h\prime(x) = \frac{ f\prime(x)g(x) + f(x)g\prime(x)}{g(x)^2}\)
  • Zanjir qoidasi: \(h(x) = f(g(x)) \textrm{ keyin } h\prime(x) = f\prime(g(x))g\prime(x)\)

Eslatma: Tangens chiziqning bir nuqtadagi qiyaligi uning shu nuqtadagi hosilasidir. Agar (x 0 , y 0 ) nuqtadagi y = f(x) egri chiziq uchun tangens chiziq chizilgan bo‘lsa, u holda funksiya hosilasidagi nuqtani oddiygina almashtirish orqali uning qiyaligi (m) olinadi.

4-misol: 10x 5

\(y\prime = \frac{dy}{dx} = \frac{d(10x^5)}{dx}\)

\(10 \times 5 x^4 = 50 x^4\) (quvvat qoidasini qo'llash)

5-misol: tan 2 x farqlang

\(y\prime = \frac{dy}{dx} = \frac{d(tan^2x)}{dx}\)

\(2\tan x^{2-1} \times \frac{d(\tan x)}{dx}\) (zanjir qoidasini qo'llash)

\(2\tan x⋅ \sec ^2 x\)

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