3582444448

• Pochodna sumy:

(/ (x) + g (*))' = ]_im /(* + ft)+g(* + ft)-(/(*) + g(*)) _{= Hm} f(* + h)-f(x) ₊

lim

h->0

h—*0

9(x-ł-/i) -<?(x)

= /' (*) + 9 (x)

h-*0

Pochodna różnicy:

(/ (x) - g (x))' - Hm /(* + J0-g(* + ft)-(/(*)-g(*)) ₌ ,;_m /(x + A)-/(x)

/»-* 0    ^L    . _    Ł

^(x-ł-/l) - </(x)

k-*0

lim

/i-»0

= /' (x) - s' (x)

Pochodna iloczynu:

(/ (x) . g (x))' = Hm /(^*)g(^*)-/WgW =

= lim /»-»o

h-»o    h

f(x-\- h) g(x + h)- f (x) g (x) + / (x) g (x + k) - f (x) £ (x + h)

_ _Hm / (*) (g (g + /Q - g (x)) + 9 (x + /i) (/ (x + h) - f (x) _    , V _lim g(x + /Q -g(x) _[

h—*0    h    j \ > _h__tQ    £

9 (x) • lim ^ ⁺ ^= / (x) • 9' (x) + /' (x) • 3 (x)

• Pochodna ilorazu:

//(x)V _ _]im {{l+h) fifł _ _Um /(x + h)g(x) - f(x)g(x + h) _

W(x)/

= lim

h—*0

= lim

h-»0

/»—»0    /l    /V—>6    /z. • 9 (x) 9 (x + /i)

f(x + h)g(x) — f (x) g (x + h) + f (x) g (x) - f (x) g (x) _

h -g{x)g (x)

g (x) (/ (x + h) — / (x)) - / (x) (g (x + h) - g (x)) _ g (x)    / (x + h) - f (x)

h- (g(x)y

(g W)'

lim

h-*o

/(x)    g(x + H)-_ff(x) /' (x) g (x) - / (x) g' (x)

lim-:- = -7-—-, gdyg (x) f- 0

h-*0

(9 (x)²)