3582525253

Proste pochodne

Wzory:	Przykłady:
(c)' = o	(2)' = 0 (100)' = 0
(ax)' = a	(*)' = ! (3ar)' = 3
(x^n)' = nx^n~^1	(a;^3)' = 3a:^2(x^5y = 5x^4
(2)'—4	(1)' =__L V x ) X^2 (-)' = --% \x) x^z
<D II CD
(a^x)^7 = a^x Ina
(lnx)^/ = — X ^(,0e°^Z>' = .!„«	(log2 x)’ = -

Oogó*)' = jh

Wzory:

{f+9)' = f' + 9' U-9)' = ? -9' (c • /)' = c • /'

(f-gy = f^,g + fsf

Działania na pochodnych Przykłady:

(a:² + a;³)' = (x²)' + (a:³)' = 2x + 3x² (x⁴ — x)' — (x⁴)' — (x)' — 4x³ — 1 (5a:³)⁷ = 5 • (a;³)⁷ = 5 • 3x² = 15a;² {x²\fx)' = (x²)'\/x + x²{\Jx)' =

= 2x\[x + a:²

_x2

= 2xy/x + —~j=

2\Jx

f'9-fg'

\g) g²

V (a:²y^-x²(^y \Vx)    (Vx)²

_ 2a; \fx — x² _

X

Pochodna funkcji złożonej

[/(»)]' = /' V

((2® + l)³)' = (y³y = 3y³-¹ ■ y' = 3y² ■ y' =

= 3(2a: + l)² • 2 = 6(2a; + l)² gdzie y = 2x + 1, y' = 2

=    _; ¹    ■ (3x^b — 2) = f-²

2 \/a;³ — 2a;    2\/a;³ — 2a;

gdzie y = x³ — 2x, y = 3a;² — 2

(e*²)' = (e*⁷)' = e^w • y' = e^x2 • 2a: = 2a:e^x2 gdzie y = x², y' = 2x

Zadania + Rozwiązania

(sina;)' = cos a;

Pochodne funkcji trygonometrycznych 1

(cos a;)' = — sina;

(arc sin a:)⁷ =    ------

(arc cos a:)⁷ = -

\/l — x²

(tg *)'

(ctg x)' = --

(arc tg x)' — (arc ctg x)' =

tg x + 1

1

= —(ctg² x + 1)

1

1 + X² -1

1 + X²