Rozważmy funkcję
![](data:image/png;base64,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)
, gdzie
![](data:image/png;base64,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)
. Zweryfikujmy czy podana
funkcja w każdym punkcie swojej dziedziny ma pochodną.
Dla dowolnej liczby x0 z przedziału
![](data:image/png;base64,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)
istnieje takie otoczeniu U(x0), że U(x0)
![](data:image/png;base64,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)
. Niech h będzie liczbą
rzeczywistą, dla której (x0+h)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAATBAMAAACw8jreAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAOgBmOjoAOjpmZgAAZjoAZrb/kDo6kNv/tmYAttv/27aQ/9uQ//+2///bDCP/0gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAALUlEQVQYV2NgIAZcExQsYOD1BinlKgCRPyaAyKeCIHEIGyLOqw027jBIHAcAAM8XCNoOnVTjAAAAAElFTkSuQmCC)
U(x0):
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Więc f’(x) =
![](data:image/png;base64,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)
, gdzie
![](data:image/png;base64,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)
.
Wniosek: Dla dowolnej liczby x0 z przedziału
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAATCAMAAAAOJhUTAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgBmOjo6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjqQZma2ZrbbZrb/kDoAkDo6kGYAkGZmkLaQkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm29vb2////7Zm/9uQ/9u2//+2///bn/so4AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA7ElEQVQ4T81SyRaCMAwsKCJuuCAugEtVCvT//88kLVKhTzl4MIe+vnYmmUnC2L+GPM/zXtq4xsld1AsPoGJyRShf4HnzHe97JYGYapkAQbiJjEdGLfXcDhlvAUupU0ADq0HYGSwFQYhlVQi3wocEddSMcuUML6h5Rpq5Bwy0UYUApqPFkHHEysPJjR57sivsDBk7KtykmEJqKk43xfioinB8AE2t1shAVeSc+iTww6Lq7h43uR4C5lcWuepuY8VwDi0UQUDOqbuanDnYjS7jfSJqFOaWFGNDWHeAekuYsYmZbdAvZr2JlmX47dMTouEWOTxMreAAAAAASUVORK5CYII=)
istnieje pochodna funkcji funkcję
![](data:image/png;base64,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)
i ma ona wzór f’(x) =
![](data:image/png;base64,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)
, dziedziną której jest
również przedział
![](data:image/png;base64,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)
.
Dziedzina funkcji
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAVCAMAAAB8FU7dAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjpmOmaQOpDbZgAAZpC2ZrbbZrb/kDoAkDpmkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ//+2///bIKgY1QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAhklEQVQoU7VO0RLCMAiDTkVXda4TtbT//50Cnjfmu3loj5CQAPwbjIrd+RVyWk5zL3gIVMXhDkJpXjk2Qct4XamCI2xVdgrA3QDsn5C9BfcaWfFy06HaIOSn2KW6H+FJdk9t7nuQFk3HxbZ9iuW8gFAo8qnEsa4zfbLUDYQsI6Ll06/ou34Dw2IFh01vyYYAAAAASUVORK5CYII=)
i dziedzina funkcji pochodnej
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAAVCAMAAACeyVWkAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjpmOmaQOpDbZgAAZgBmZma2ZpC2ZrbbZrb/kDoAkDpmkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ//+2///bO+k5vAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAkElEQVQoU72PyQ4CMQxDk7IEKNswhQBd/v8ziacSIgNnfGlrOS8u0Z+lbFoen35tjWFsidfezby4UZEwOlsRq5HPzk28o68ssEQThkj7gRQuiVdWIvPp0t2Md5EJqwhkzAL7ELBtHgC11F3sD2FzxVgbUDr5MsYxo+47+S0FsRy82Qb0mKsIVs5U4/ZH9CP0AjwJBufge4HTAAAAAElFTkSuQmCC)
nie muszą być równe, ale zawsze zachodzi
zależność
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAVCAMAAAB8FU7dAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjpmOmaQOpDbZgAAZpC2ZrbbZrb/kDoAkDpmkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ//+2///bIKgY1QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAhklEQVQoU7VO0RLCMAiDTkVXda4TtbT//50Cnjfmu3loj5CQAPwbjIrd+RVyWk5zL3gIVMXhDkJpXjk2Qct4XamCI2xVdgrA3QDsn5C9BfcaWfFy06HaIOSn2KW6H+FJdk9t7nuQFk3HxbZ9iuW8gFAo8qnEsa4zfbLUDYQsI6Ll06/ou34Dw2IFh01vyYYAAAAASUVORK5CYII=)
.
Definicja 1
Niech f będzie dowolną funkcją określoną w zbiorze
. Funkcją pochodną funkcji f
(lub pochodną funkcji f) nazywamy funkcję, która każdej liczbie x0 z dziedziny
przyporządkowuje liczbę f’(x0) – o ile f’(x0)
istnieje – i oznaczamy ją f’. Zbiór tych liczb x0, w których funkcja f jest
różniczkowalna, stanowi dziedzinę funkcji f’, która oznaczamy
.
Pochodną funkcji oznaczamy również w następujący sposób:
lub
.
Przy okazji pochodnych należy zaznaczyć że pochodna funkcji
w punkcie nie jest tym samym co pochodna funkcji. Pochodna funkcji w punkcie
jest niczym innym jak liczbą (granicą ilorazu różnicowego), natomiast pochodna
funkcji jest to funkcja, która argumentom x przypisuje liczbę równą pochodnej
funkcji w punkcie x.
Funkcja
|
Pochodna
funkcji
|
Dziedzina
pochodnej
|
f(x) = c
|
f’(x) = 0
|
(c – dowolna liczba rzeczywista, f – funkcja
stała)
|
f(x) = ax+b
|
f’(x) = a
|
(a, b – dowolne liczby rzeczywiste)
|
f(x) = a +bx+c
|
f’(x) = 2ax+b
|
(a, b, c – dowolne liczby
rzeczywiste)
|
f(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAATCAMAAACuuX39AAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAA6ADpmADqQAGa2OgAAOgA6Oma2ZgA6ZjoAZjo6ZjqQZmaQZrbbZrb/kDoAkDpmkGaQkNvbkNv/tmYAtmaQtpA6tpC2tv//25A627Zm27a229uQ29u22////7Zm/9uQ/9u2//+2///b3K/0GgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZklEQVQoU7WPyQ6AIAxER0XEDRVx3+H//9GamIh3fZcmzWsnA/zKEbMxDNYnY5p5blJngYFhi0jYyTMxA7SEzjvYahGq7AEjarReTYot/Gs40PGLPXHDgCObuWroyY32e1t48tPOJ+9QBLzJdllHAAAAAElFTkSuQmCC)
|
f’(x) = ![](data:image/png;base64,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)
|
(n – liczba naturalna większa od 1)
|
f(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAATCAMAAACuuX39AAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjqQOma2OpDbZgA6ZgBmZjoAZjo6ZmaQZrb/kDoAkDpmkGZmkGaQkLbbkNvbkNv/tmYAtmaQtv//25A629uQ29u22////7Zm/9uQ/9u2//+2///bvPaHgwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQoU6WPRxaAIBBDQcEGNiyIDSn3P6NtIbB1lpn8lwSAPyeiycdNufiCKiyDraOtRPej6xEwfhiVYmmqBBjKGdES2GGnvLteKpcHJLfFMhTkrUkQV2PpKrrZMj5/nQRaLIvcPn/mv+wJuGYFzXIwRGQAAAAASUVORK5CYII=)
|
f’(x) = k![](data:image/png;base64,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)
|
(k – liczba całkowita ujemna)
|
f![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAATCAMAAADyMz7ZAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgAAOgA6OgBmOjpmOjqQOma2OpC2OpDbZgA6ZjoAZjo6ZmaQZrbbZrb/kDoAkDo6kDpmkGY6kGaQkNvbkNv/tmYAtmaQtpBmtrbbttvbtv//25A627Zm29uQ29u22////7Zm/9uQ/9u2//+2///bJpq8jQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA20lEQVQ4T9WSWxOCIBCFEe1eWkYXs4tURO7//4FBLGg2OI7TS7zIjN/Zc3ZZQv7qwGkmMDB3tw4dwCatKDkpOkgMwhd19BFZd18B2AVUQ2V8qCPA1u2WwOaCDxTTtMg/AnwXkVNBcl0812oih5EoE33lnqyWUEbX8eWshNoB9vc4277nggmABeZQ04kjyoSmLFQoRgMWGsI7HUdgfBNVRbRf71gtgUJ0kEsUYKFmVDUGS6DQPMdzdRtlR5Xc+xyOcIPWC5CHBTCqh+xrsSKcsPfKkb5L3r5hP//7Atu7EjoA2wPCAAAAAElFTkSuQmCC)
|
f’(x) = a![](data:image/png;base64,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)
|
(a – dowolna liczba rzeczywista)
|
f(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAaCAMAAAC5Oj0UAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6OgAAOgBmOjqQOmZmOmaQOpDbZgA6ZmZmZpC2Zrb/kDoAkNvbkNv/tmYAtmZmtpCQtraQtrbbtrb/ttuQtv//25A627Zm27aQ2////7Zm/9uQ//+2///bQEdC3wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAUElEQVQoU2NgwAEkWcQhMvKCwjAmA4Ms8UwBRkZGDlym4xQHagIBkvUR1CDBxC/DzANSJicqwcYtxQ/RIcMqDWEIiXFx8omAmbzs0gJMYHkAobADV/9ALdcAAAAASUVORK5CYII=)
|
f’(x) = ![](data:image/png;base64,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)
|
![](data:image/png;base64,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)
|
![](data:image/png;base64,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)
|
![](data:image/png;base64,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)
|
![](data:image/png;base64,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)
|
Twierdzenie 1Jeśli funkcje f i g są różniczkowalne w zbiorze D, to dla
dowolnej liczby x
:
I. [
’ =
(wzór na pochodną sumy funkcji)
II. [
’ =
(wzór na pochodną różnicy funkcji)
III. [
(wzór na pochodną iloczynu funkcji)
IV.
gdzie
c – dowolna liczba rzeczywista
V.
przy dodatkowym założeniu, że g(x)
w zbiorze D (wzór na pochodną ilorazu
funkcji).
Przykład 1
Oblicz pochodną funkcji:
a) ![](data:image/png;base64,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)
b) ![](data:image/png;base64,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)
c) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAAATCAMAAACwXI8LAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmaQZpBmZpC2ZrbbZrb/kDoAkDo6kDpmkDqQkGY6kGaQkLb/kNvbkNv/tmYAtmY6tmaQtpBmttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2/9vb//+2///bkC+8TgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACDUlEQVRIS+1Wa1PCMBBMQKCoUPCFVVEoSosPWmz+/28zyV2SXvqagW9qZmBaunu3t3dJYex//WYHxNtkj/Wl9qqm4BKOsfeAXxpWDdYGIqQuFyvpxcPCcfLzpCkAxV3sD+F9Sy4MREhd0hhz6fM5729ZOi1zsmGTGxTHmHisKUNEXC2pGgJp0vcm4IN1mzKx5HywlQiTvggXxXVSzAhLRA1ueDjGdg7onoknqShVsnQg/UBEU/mp1lxiRaO9iC1L1aThvlExsdEV6+N2L+4Z1a20yKUCOVLar7jsWPlYlqC/NEtWIr2XF/FIBcqD4b6Yq0tQXF0eTrqWWeOotgxkqEBAMncuif7J01bo+YX02D0Q+vw1W8H8mFJxcjjvQc8pLg9gqmBRbShIBbJNUIlLSTxWEcrz4SNQ8SA9BkS6iHBamzaDjzO6/BqwN5DFaMPtapNA19SCyrOAn92WtKH1xvYU7W/vqTTdtsn1nfimZ5r2VGxwiAnZ212gB9IjDn3KrzCk0er74eOcMtpTGBs1BLIIJBlpNknNJIhIF63T427C1h5uPserVym84wyxuLI0Mm/Gd3eGsGwi0Zs188jEt8OytAlsfep4jAeJiHpuFklmvKG4Jm2m5NLZW4RqsPqJS1LdQXHvTk8CGI3jJp065p1VJ578dso7S7ij56h3fYe4k971Ydu/iE5X/izgBxWnR8FWQzuFAAAAAElFTkSuQmCC)
d) ![](data:image/png;base64,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)
Na mocy twierdzenia 1 jesteśmy w łatwy sposób policzyć
każdą z wymienionych pochodnych:
Ad. a)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Ad. b)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Ad. c)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
[
’ =
=
= ![](data:image/png;base64,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)
Ad. d)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=
= ![](data:image/png;base64,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)
Zadania do zrobienia
1. Wyznacz pochodną funkcji:
a)
b)
c)
d)
Odp. a)
b)
c)
d)
.
2. Wyznacz pochodną funkcji:
a)
b)
c)
Odp. a)
b)
c)
3. Zbadaj, czy istnieją takie wartości
parametrów
,
, dla których funkcja
jest różniczkowana w zbiorze R. Wyznacz
.
.
Odp.
.