Hosila hisoblash qoidalari

Bu nima va nega kerak

Oldingi darsda biz hosilani ta'rif bo'yicha limit hisoblash orqali topdik. Ammo har bir ifoda uchun limit hisoblash ko'p vaqt oladi.

Differensial hisobning eng qulay tomoni shundaki, asosiy amallar va elementar funksiyalar hosilalari jadvalidan foydalanib, istalgan murakkab ifodaning hosilasini bir necha soniyada topish mumkin.

Ushbu darsda hosila hisoblashning 5 ta asosiy qoidasi, hosilalar jadvali va ularni amalda qo'llashni o'rganamiz.

Asosiy g'oya

1. Arifmetik amallarning hosila qoidalari

Aytaylik, $u = u(x)$ va $v = v(x)$ differensiyallanuvchi funksiyalar, $c$ esa o'zgarmas son bo'lsin:

1. O'zgarmas ko'paytuvchini chiqarish: $(c \cdot u)' = c \cdot u'$

2. Yig'indi va ayirmaning hosilasi: $(u \pm v)' = u' \pm v'$

3. Ko'paytmaning hosilasi: $(u \cdot v)' = u'v + uv'$

4. Bo'linmaning (kasrning) hosilasi: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \quad (v \neq 0)$

5. Murakkab funksiya hosilasi: Agar $y = f(g(x))$ bo'lsa: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ Xususan: $(f(kx + b))' = k \cdot f'(kx + b)$.

2. Asosiy hosilalar jadvali

Funksiya $f(x)$	Hosilasi $f'(x)$	Misol
$C$ (o'zgarmas)	$0$	$(7)' = 0$
$x$	$1$	$(x)' = 1$
$x^n$	$n x^{n-1}$	$(x^5)' = 5x^4$
$\sqrt{x}$	$\dfrac{1}{2\sqrt{x}}$	$\left(x^{1/2}\right)' = \frac{1}{2\sqrt{x}}$
$\dfrac{1}{x}$	$-\dfrac{1}{x^2}$	$\left(x^{-1}\right)' = -\frac{1}{x^2}$
$\sin x$	$\cos x$	$(\sin(3x))' = 3\cos(3x)$
$\cos x$	$-\sin x$	$(\cos(4x))' = -4\sin(4x)$
$\text{tg}\,x$	$\dfrac{1}{\cos^2 x}$	$x \neq \frac{\pi}{2} + \pi k$
$\text{ctg}\,x$	$-\dfrac{1}{\sin^2 x}$	$x \neq \pi k$
$e^x$	$e^x$	$(e^{3x})' = 3e^{3x}$
$a^x$	$a^x \ln a$	$(2^x)' = 2^x \ln 2$
$\ln x$	$\dfrac{1}{x}$	$x > 0$
$\log_a x$	$\dfrac{1}{x \ln a}$	$x > 0, a > 0, a \neq 1$

Yechilgan misollar

1-misol. Ko'phad va darajali funksiyalar hosilasi

$f(x) = 2x^4 - 5x^3 + 4x - 9$ funksiyaning $x_0 = 1$ nuqtadagi hosilasi $f'(1)$ ni hisoblang.

Yechish: Hosilalar jadvali va amallar qoidasini qo'llaymiz: $f'(x) = 2(4x^3) - 5(3x^2) + 4(1) - 0 = 8x^3 - 15x^2 + 4.$ $x = 1$ da: $f'(1) = 8(1)^3 - 15(1)^2 + 4 = 8 - 15 + 4 = -3.$

Javob: $-3$.

2-misol. Ko'paytmaning hosilasi

$f(x) = (2x - 1)(x^2 + 3)$ funksiyaning $x_0 = 2$ dagi hosilasi $f'(2)$ ni hisoblang.

Yechish: $(uv)' = u'v + uv'$ formuladan: $u = 2x - 1 \implies u' = 2$; $v = x^2 + 3 \implies v' = 2x$. $f'(x) = 2(x^2 + 3) + (2x - 1)(2x) = 2x^2 + 6 + 4x^2 - 2x = 6x^2 - 2x + 6.$ $x = 2$ da: $f'(2) = 6(4) - 2(2) + 6 = 24 - 4 + 6 = 26.$

Javob: $26$.

3-misol. Kasrning hosilasi

$f(x) = \dfrac{3x - 1}{x + 2}$ funksiya berilgan. $f'(1)$ ni toping.

Yechish: Bo'linma qoidasidan foydalanamiz: $f'(x) = \frac{(3x - 1)'(x + 2) - (3x - 1)(x + 2)'}{(x + 2)^2} = \frac{3(x + 2) - (3x - 1) \cdot 1}{(x + 2)^2} = \frac{7}{(x + 2)^2}.$ $x = 1$ da: $f'(1) = \frac{7}{(1 + 2)^2} = \frac{7}{9}.$

Javob: $\dfrac{7}{9}$.

4-misol. Murakkab funksiyaning hosilasi

Quyidagi funksiyalarning hosilalarini hisoblang: a) $f(x) = (3x - 5)^4$, $f'(2)$ ni toping; b) $g(x) = \sin(2x) - 3\cos x$, $g'(0)$ ni toping.

Yechish: a) $f'(x) = 4(3x - 5)^3 \cdot (3x - 5)' = 4(3x - 5)^3 \cdot 3 = 12(3x - 5)^3$. $x = 2$ da: $f'(2) = 12(6 - 5)^3 = 12$. b) $g'(x) = 2\cos(2x) + 3\sin x$. $x = 0$ da: $g'(0) = 2\cos 0 + 3\sin 0 = 2(1) + 0 = 2$.

Javob: a) $12$; b) $2$.

Mashq

1. $f(x) = 5x^3 - 4x^2 + 6x - 1$ funksiyaning $x_0 = 1$ nuqtadagi hosilasi $f'(1)$ ni toping.
2. $f(x) = \sqrt{x} + \dfrac{1}{x}$ funksiyaning $x_0 = 4$ nuqtadagi hosilasini hisoblang.
3. $f(x) = (x^2 + 1)(2x - 3)$ funksiyaning $x_0 = 2$ dagi hosilasi $f'(2)$ ni toping.
4. $f(x) = \dfrac{2x - 1}{x + 1}$ funksiyaning $x_0 = 1$ dagi hosilasi $f'(1)$ ni hisoblang.
5. $f(x) = (2x + 1)^4$ funksiyaning $x_0 = 0$ dagi hosilasini toping.
6. $f(x) = \sin(3x) + \cos(2x)$ funksiyaning $x_0 = 0$ dagi hosilasini toping.
7. $f(x) = x \cdot e^x$ funksiyaning $x_0 = 0$ nuqtadagi hosilasi $f'(0)$ ni toping.
8. $f(x) = \ln(2x - 1)$ funksiyaning $x_0 = 1$ dagi hosilasini hisoblang.
9. $f(x) = \dfrac{x^2}{x - 2}$ funksiyaning $x_0 = 4$ dagi hosilasi $f'(4)$ ni toping.
10. $f(x) = \sqrt{2x + 5}$ funksiyaning $x_0 = 2$ dagi hosilasini hisoblang.

Javoblar va yechim yo'llari

1. $f'(x) = 15x^2 - 8x + 6 \implies f'(1) = 15 - 8 + 6 = 13$. Javob: $13$.

2. $f'(x) = \dfrac{1}{2\sqrt{x}} - \dfrac{1}{x^2} \implies f'(4) = \dfrac{1}{4} - \dfrac{1}{16} = \dfrac{3}{16}$. Javob: $\dfrac{3}{16}$.

3. $f'(x) = 2x(2x - 3) + 2(x^2 + 1) = 6x^2 - 6x + 2 \implies f'(2) = 14$. Javob: $14$.

4. $f'(x) = \dfrac{2(x + 1) - (2x - 1)}{(x + 1)^2} = \dfrac{3}{(x + 1)^2} \implies f'(1) = \dfrac{3}{4}$. Javob: $\dfrac{3}{4}$.

5. $f'(x) = 4(2x + 1)^3 \cdot 2 = 8(2x + 1)^3 \implies f'(0) = 8$. Javob: $8$.

6. $f'(x) = 3\cos(3x) - 2\sin(2x) \implies f'(0) = 3(1) - 0 = 3$. Javob: $3$.

7. $f'(x) = 1 \cdot e^x + x \cdot e^x = e^x(1 + x) \implies f'(0) = 1$. Javob: $1$.

8. $f'(x) = \dfrac{(2x - 1)'}{2x - 1} = \dfrac{2}{2x - 1} \implies f'(1) = 2$. Javob: $2$.

9. $f'(x) = \dfrac{2x(x - 2) - x^2}{(x - 2)^2} = \dfrac{x^2 - 4x}{(x - 2)^2} \implies f'(4) = 0$. Javob: $0$.

10. $f'(x) = \dfrac{2}{2\sqrt{2x + 5}} = \dfrac{1}{\sqrt{2x + 5}} \implies f'(2) = \dfrac{1}{3}$. Javob: $\dfrac{1}{3}$.

Keng tarqalgan xatolar

1. Ko'paytma hosilasini oddiy hosilalar ko'paytmasi deb yozish. ✗ $(u \cdot v)' = u' \cdot v'$ deb hisoblash (masalan $(x \sin x)' = 1 \cdot \cos x$). ✓ Ko'paytma qoidasi: $(uv)' = u'v + uv'$, ya'ni $(x \sin x)' = \sin x + x \cos x$.

2. Kasr hosilasida suratdagi ishorani va maxraj kvadratini adashtirish. ✗ $\left(\dfrac{u}{v}\right)' = \dfrac{u'v + uv'}{v}$ deb yozish. ✓ Bo'linma qoidasida suratda ayirma bo'ladi va maxraj kvadratga ko'tariladi: $\dfrac{u'v - uv'}{v^2}$.

3. Murakkab funksiyada ichki hosilani unutish. ✗ $(\sin(3x))' = \cos(3x)$ deb yakunlash. ✓ Ichki ifodaning hosilasiga ham ko'paytiriladi: $(\sin(3x))' = 3\cos(3x)$.

4. Ko'rsatkichli funksiya hosilasini darajali funksiya deb hisoblash. ✗ $(3^x)' = x \cdot 3^{x-1}$ deb yozish. ✓ Asosi o'zgarmas son bo'lsa: $(a^x)' = a^x \ln a$, ya'ni $(3^x)' = 3^x \ln 3$.

DTM uslubidagi test

Ushbu mavzu bo'yicha tayyorlangan 5 ta test savoli platformada interaktiv tarzda ishlaydi: o'z javobingizni tanlang va to'liq yechimni ko'ring.

Keyingi qadam

Biz asosiy amallar va elementar funksiyalar hosilalarini hisoblashni o'rgandik. Keyingi darsda hosilaning eng muhim geometrik qo'llanilishi bo'lgan Urinma tenglamasi va unga oid DTM masalalarini ko'rib chiqamiz.