Murakkab funksiyaning hosilasi: matematika darsi

$(3x-1)^4$ ning hosilasini qavslarni ochib ham topish mumkin, lekin $(3x-1)^{40}$ da bu juda uzun ish. Murakkab funksiya qoidasi bilan bir qatorda yoziladi: $40(3x-1)^{39}\cdot 3$ . Ildiz, logarifm, sinus ichida ifoda turgan barcha masalalarda shu qoida ishlatiladi.

Murakkab funksiya nima

$y=f(g(x))$ ko'rinishdagi funksiya murakkab funksiya deyiladi: avval $g(x)$ hisoblanadi, natija $f$ ga beriladi. $g$ ichki, $f$ tashqi funksiya. Masalan:

Funksiya	Ichki $u$	Tashqi
Funksiya $(3x-1)^4$	Ichki $u$ $u=3x-1$	Tashqi $u^4$
Funksiya $\sqrt{x^2+16}$	Ichki $u$ $u=x^2+16$	Tashqi $\sqrt{u}$
Funksiya $\sin 5x$	Ichki $u$ $u=5x$	Tashqi $\sin u$
Funksiya $\sin^2 x=(\sin x)^2$	Ichki $u$ $u=\sin x$	Tashqi $u^2$
Funksiya $e^{x^2-3x}$	Ichki $u$ $u=x^2-3x$	Tashqi $e^u$
Funksiya $\ln\cos x$	Ichki $u$ $u=\cos x$	Tashqi $\ln u$

Asosiy qoida

\big(f(g(x))\big)'=f'(g(x))\cdot g'(x)

g

nuqta

x

da,

f

esa

g(x)

nuqtada differensiallanuvchi bo'lsa

Avval tashqi funksiyaning hosilasini ichki ifodani o'zgartirmasdan yozamiz, keyin ichki ifodaning hosilasiga ko'paytiramiz.

Sababi: $\dfrac{\Delta f}{\Delta x}=\dfrac{\Delta f}{\Delta u}\cdot\dfrac{\Delta u}{\Delta x}$ , $\Delta x\to 0$ da $\Delta u\to 0$ ham bo'ladi. Misol:

\big((3x-1)^4\big)'=4(3x-1)^3\cdot 3

=12(3x-1)^3

Ochilgan ko'rinish bilan tekshirsak,

x=1

da ikkalasi ham

96

beradi

Ichki funksiya bilan jadval

Funksiya	Hosila	Shart
Funksiya $u^n$	Hosila $nu^{n-1}\cdot u'$	Shart $n$ natural; butun manfiy $n$ da $u\ne 0$
Funksiya $\dfrac{1}{u}$	Hosila $-\dfrac{u'}{u^2}$	Shart $u\ne 0$
Funksiya $\sqrt{u}$	Hosila $\dfrac{u'}{2\sqrt{u}}$	Shart $u>0$
Funksiya $\sin u$	Hosila $u'\cos u$	Shart
Funksiya $\cos u$	Hosila $-u'\sin u$	Shart
Funksiya $\mathrm{tg}\,u$	Hosila $\dfrac{u'}{\cos^2 u}$	Shart $\cos u\ne 0$
Funksiya $\mathrm{ctg}\,u$	Hosila $-\dfrac{u'}{\sin^2 u}$	Shart $\sin u\ne 0$
Funksiya $e^u$	Hosila $u'e^u$	Shart
Funksiya $a^u$	Hosila $u'a^u\ln a$	Shart $a>0$ , $a\ne 1$
Funksiya $\ln u$	Hosila $\dfrac{u'}{u}$	Shart $u>0$
Funksiya $\log_a u$	Hosila $\dfrac{u'}{u\ln a}$	Shart $u>0$ ; $a>0$ , $a\ne 1$
Funksiya $\arcsin u$	Hosila $\dfrac{u'}{\sqrt{1-u^2}}$	Shart $\|u\|<1$
Funksiya $\arccos u$	Hosila $-\dfrac{u'}{\sqrt{1-u^2}}$	Shart $\|u\|<1$
Funksiya $\mathrm{arctg}\,u$	Hosila $\dfrac{u'}{1+u^2}$	Shart

\big(f(kx+b)\big)'=k\,f'(kx+b)

Ichki funksiya chiziqli bo'lsa, hosila shunchaki

k

ga ko'payadi

Funksiya	Hosila
$\sin 5x$	$5\cos 5x$
$e^{-2x}$	$-2e^{-2x}$
$\sqrt{x^2+16}$	$\dfrac{x}{\sqrt{x^2+16}}$
$\ln(x^2+1)$	$\dfrac{2x}{x^2+1}$
$\mathrm{arctg}\,2x$	$\dfrac{2}{1+4x^2}$
$\cos(x^3+1)$	$-3x^2\sin(x^3+1)$

Ildizli qatorda avval $\dfrac{2x}{2\sqrt{x^2+16}}$ chiqadi, keyin $2$ qisqaradi.

Bir necha qavatli funksiya

Tashqaridan ichkariga qavatma-qavat yuramiz va har bir qavatning hosilasini ko'paytiramiz:

u=g(h(x))

\big(f(u)\big)'=f'(u)\cdot g'(h(x))\cdot h'(x)

$\sin^2 x$ da tashqi funksiya kvadrat:

(\sin^2 x)'=2\sin x\cos x=\sin 2x

(\cos^2 x)'=-\sin 2x

Uch qavat bo'lsa ham shunday:

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