📐 Differensial Hisob — Nazariya

Kirish

Differensial hisob — o'zgarishlarni o'rganuvchi matematika bo'limi. Robotika, dronlar va raketalar uchun tezlik, tezlanish, optimizatsiya va boshqarish tizimlarida asosiy vosita.

1. Limit (Chegara)

Ta'rif

$x$ $a$ ga yaqinlashganda $f(x)$ ning limiti $L$ ga teng:

$$\lim_{x \to a} f(x) = L$$

Bu degani, $x$ $a$ ga qanchalik yaqin bo'lsa, $f(x)$ $L$ ga shunchalik yaqin bo'ladi.

Asosiy limitlar

Limit	Qiymat
$\lim_{x \to 0} \frac{\sin x}{x}$	$1$
$\lim_{x \to 0} \frac{1 - \cos x}{x}$	$0$
$\lim_{x \to 0} \frac{e^x - 1}{x}$	$1$
$\lim_{x \to \infty} (1 + \frac{1}{x})^x$	$e$
$\lim_{x \to 0} \frac{\ln(1+x)}{x}$	$1$

Limit qoidalari

1. Yig'indi: $\lim[f(x) + g(x)] = \lim f(x) + \lim g(x)$
2. Ko'paytma: $\lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)$
3. Bo'linma: $\lim\frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$, agar $\lim g(x) \neq 0$

L'Hôpital qoidasi

Agar $\frac{0}{0}$ yoki $\frac{\infty}{\infty}$ noaniqlik bo'lsa:

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

Misol

$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$

2. Hosila (Derivative)

Ta'rif

Funksiyaning $x = a$ nuqtadagi hosilasi:

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

Geometrik ma'no

Hosila — funksiya grafigiga $x = a$ nuqtada o'tkazilgan urinma to'g'ri chiziqning burchak koeffitsienti.

       y
       |     /
       |    /  ← urinma (slope = f'(a))
       |   / 
       |  *  ← (a, f(a))
       | /
       |/
       +----------→ x
           a

Fizik ma'no

• Tezlik: $v(t) = \frac{ds}{dt} = s'(t)$ — pozitsiyaning vaqt bo'yicha hosilasi
• Tezlanish: $a(t) = \frac{dv}{dt} = v'(t) = s''(t)$ — tezlikning hosilasi

3. Hosila qoidalari

Asosiy hosilalar

Funksiya $f(x)$	Hosila $f'(x)$
$c$ (konstanta)	$0$
$x^n$	$nx^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\frac{1}{x}$
$\log_a x$	$\frac{1}{x \ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x = \frac{1}{\cos^2 x}$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$
$\arccos x$	$-\frac{1}{\sqrt{1-x^2}}$
$\arctan x$	$\frac{1}{1+x^2}$

Hosila qoidalari

1. Konstanta ko'paytma: $[cf(x)]' = cf'(x)$

2. Yig'indi/Ayirma: $[f(x) \pm g(x)]' = f'(x) \pm g'(x)$

3. Ko'paytma (Leibniz): $[f(x) \cdot g(x)]' = f'(x)g(x) + f(x)g'(x)$

4. Bo'linma: $\left[\frac{f(x)}{g(x)}\right]' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

5. Zanjir qoidasi (Chain Rule): $[f(g(x))]' = f'(g(x)) \cdot g'(x)$

Zanjir qoidasi misol

$y = \sin(x^2)$ uchun:

• Tashqi funksiya: $\sin(u)$, hosilasi: $\cos(u)$
• Ichki funksiya: $u = x^2$, hosilasi: $2x$

$y' = \cos(x^2) \cdot 2x = 2x\cos(x^2)$

4. Yuqori tartibli hosilalar

Ikkinchi hosila: $f''(x) = \frac{d^2f}{dx^2} = \frac{d}{dx}\left(\frac{df}{dx}\right)$

$n$-tartibli hosila: $f^{(n)}(x) = \frac{d^nf}{dx^n}$

Fizik ma'no

Tartib	Mexanik ma'no
$s(t)$	Pozitsiya
$s'(t) = v(t)$	Tezlik
$s''(t) = a(t)$	Tezlanish
$s'''(t) = j(t)$	Jerk (silkinish)
$s^{(4)}(t)$	Snap

Robotikada jerk

Robot harakatlarida jerk ($j = \frac{da}{dt}$) cheklash muhim — bu mexanik stress va tebranishlarni kamaytiradi.

5. Qisman hosilalar (Partial Derivatives)

Ko'p o'zgaruvchili funksiyalar uchun:

$f(x, y) \quad \Rightarrow \quad \frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}$

Ta'rif

$x$ bo'yicha qisman hosila ($y$ ni konstanta deb):

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

Gradient

Gradient — barcha qisman hosilalardan tuzilgan vektor:

$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$

Gradient funksiya eng tez o'sish yo'nalishini ko'rsatadi.

Misol

$f(x, y) = x^2 + 3xy + y^2$ uchun: $\frac{\partial f}{\partial x} = 2x + 3y$ $\frac{\partial f}{\partial y} = 3x + 2y$ $\nabla f = (2x + 3y, 3x + 2y)$

Hessian matritsasi

Ikkinchi tartibli qisman hosilalar matritsasi:

\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}$$ ## 6. Differensial Funksiyaning differensiali: $$df = f'(x) \cdot dx$$ Ko'p o'zgaruvchili: $$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$$ ### Taxminiy hisoblash Agar $\Delta x$ kichik bo'lsa: $$f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x$$ ## 7. Ekstremal qiymatlar ### Lokal ekstremumlar 1. **Kritik nuqtalar:** $f'(x) = 0$ yoki $f'(x)$ mavjud emas 2. **Ikkinchi hosila testi:** - $f''(x) > 0$ → lokal minimum - $f''(x) < 0$ → lokal maksimum - $f''(x) = 0$ → test noaniq ### Ko'p o'zgaruvchili ekstremumlar Kritik nuqta: $\nabla f = 0$ Hessian testi: - $H > 0$ va $\frac{\partial^2 f}{\partial x^2} > 0$ → minimum - $H > 0$ va $\frac{\partial^2 f}{\partial x^2} < 0$ → maksimum - $H < 0$ → egar nuqta ## 8. Lagrange ko'paytuvchilar Cheklash bilan optimizatsiya: **Masala:** $f(x, y)$ ni $g(x, y) = 0$ sharti bilan optimize qilish. **Yechish:** $$\nabla f = \lambda \nabla g$$ Bu teng qo'shiladi: $$g(x, y) = 0$$ :::tip[Misol] $f(x,y) = xy$ ni $x + y = 10$ sharti bilan maksimallash: $\nabla f = (y, x)$, $\nabla g = (1, 1)$ $(y, x) = \lambda(1, 1)$ → $y = \lambda$, $x = \lambda$ $x + y = 10$ → $2\lambda = 10$ → $\lambda = 5$ Javob: $x = y = 5$, $f_{max} = 25$ ::: ## 9. Taylor qatori Funksiyani polinom bilan taxminlash: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ ### Maclaurin qatori (a = 0) $$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$$ ### Muhim qatorlar $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$ $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$$ $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$$ $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... \quad (|x| < 1)$$ ## 10. Amaliy misollar ### Raketa tezligi Tsiolkovsky tenglamasi: $$v = v_e \ln\frac{m_0}{m}$$ Tezlanish (massa kamayishi bilan): $$a = \frac{dv}{dt} = \frac{v_e}{m}\frac{dm}{dt}$$ ### Robot manipulyator Jacobian orqali tezlik: $$\dot{x} = J(\theta)\dot{\theta}$$ bu yerda $J_{ij} = \frac{\partial x_i}{\partial \theta_j}$ ### PID kontroller $$u(t) = K_p e(t) + K_i \int e(t)dt + K_d \frac{de(t)}{dt}$$ $D$ term — xato hosilasi. ## Xulosa Differensial hisob asoslari: - ✅ Limit va uzviylik - ✅ Hosila va uning geometrik/fizik ma'nosi - ✅ Hosila qoidalari (zanjir qoidasi muhim!) - ✅ Qisman hosilalar va gradient - ✅ Ekstremal qiymatlar va optimizatsiya - ✅ Taylor qatori