Trigonometrik funksiyalar hosilasi. Teskari trigonometrik funksiyaning hosilasi. Reja

Isbot. (2) formulani tengliklarga va murakkab funksiyani differensiallash qoidalariga asoslanib isbotlaymiz.

(3) va (4) formulalarni esa bo’linmaning hosilasini toppish qoidasi va sinus funksiya hosilalarini toppish formulalaridan foydalanib isbotlaymiz.

1

Logarifmik funksiyani hosilasi (log_ax)` =

x ln x

x

Кo’rsatkichli funksiyani hosilasi (e^x)` = е^х;

Теskari trigonometric funksiyalarni hosilalari.

1 1

(arcsinx)` = ; (arctgx)` = -

√1-x² 1+x ²

1 1

(arccosx)` = - ; (arcctgx)` =

√1-x² 1+x ²

1-misol: у=(х³-1) (3х+5) ni hosilasini toping.

Yechish: у= 3х⁴-3х+5х³-5 = 3х⁴+5х³-3х-5

У`= (3х<sup>4</sup>-3х+5х<sup>3</sup>-5)` = (3х⁴)`+(5х<sup>3</sup>)`-(3х)`-(5)`= 3(х⁴)`-(3х)`+5(х³)`-(5)` = 12x³+15x²-3;

Х²

2-misol. у= ni hosilasini toping.

1+х

x² ( x²)`.(1+x)-(1+x)`.x² 2x(1+x)-x² 2x+2x²-x²

y`= ( ) ` = = = =

1+x (1+x)² (1+x)² (1+x)²

x²+2x x(x+2)

= =

(1+x)² (1+x)²

3-misol. у= 5 cos 6x ni hosilasini toping.

y`= (5cos6x)` = 5(cos6x)` = 5 (-sin6x) (6x)` = -5 sin6x.6 = -30 sin6x

4-misol. y= e^cosx ni hosilasini toping.
у` =( e<sup>cosx</sup>)` = e^cosx(cosx)` = - e^cosx.sinx
5-misol .y= arcsin 10 x hosilasini toping.
1 10

y` = (arcsin 10 x)`= .(10x)`=

√1- (10x)² √1- 100x²