Verskil tussen weergawes van "Lys van afgeleides"

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Saam met [[integrasie]] vorm [[differensiasie]] die hoofbewerkings van [[calculus]]. In die onderstaande lys is ''f'' en ''g'' differensieerbare [[funksie|funksies]]s van die [[reële]] [[getal]] ''s''. ''c'' is ook 'n reële getal.
 
Hierdie '''lys van afgeleides''' is voldoende om enige elementêre funksie te differensieer.
 
== Algemene reëls by die afleiding van funksies ==
: <math>\left({cf}\right)' = cf'</math>
: <math>\left({f + g}\right)' = f' + g'</math>
; [[Produkreël]]
: <math>\left({fg}\right)' = f'g + fg'</math>
; [[Kwosiëntreël]]
: <math>\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0</math>
; [[Kettingreël]]
: <math>(f \circ g)' = (f' \circ g)g'</math>
 
== Afgeleides van eenvoudige funksies ==
 
: <math>{d \over dx} c = 0</math>
 
: <math>{d \over dx} x = 1</math>
 
: <math>{d \over dx} cx = c</math>
 
: <math>{d \over dx} |x| = {|x| \over x} = \sgn x,\qquad x \ne 0</math>
 
: <math>{d \over dx} x^c = cx^{c-1} \qquad \mbox{met beide } x^c \mbox{ en } cx^{c-1} \mbox { gedefinieer}</math>
 
: <math>{d \over dx} \left({1 \over x}\right) = {d \over dx} \left(x^{-1}\right) = -x^{-2} = -{1 \over x^2}</math>
 
: <math>{d \over dx} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -{c \over x^{c+1}}</math>
 
: <math>{d \over dx} \sqrt{x} = {d \over dx} x^{1\over 2} = {1 \over 2} x^{-{1\over 2}} = {1 \over 2 \sqrt{x}}, \qquad x > 0</math>
 
== Afgeleides van [[eksponensiaalfunksies]] en [[logaritmes]] ==
 
: <math>{d \over dx} c^x = {c^x \ln c },\qquad c > 0</math>
 
: <math>{d \over dx} e^x = e^x</math>
 
: <math>{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1</math>
 
: <math>{d \over dx} \ln x = {1 \over x},\qquad x > 0</math>
 
: <math>{d \over dx} \ln |x| = {1 \over x}</math>
 
: <math>{d \over dx} x^x = x^x(1+\ln x)</math>
 
== Afgeleides van [[trigonometrie|trigonometriese]]se funksies ==
{|
|-
| valign="top" |
: <math>{d \over dx} \sin x = \cos x</math>
 
: <math>{d \over dx} \cos x = -\sin x</math>
 
: <math>{d \over dx} \tan x = \sec^2 x = { 1 \over \cos^2 x}</math>
 
: <math>{d \over dx} \cscsec x = -\csctan x \cotsec x</math>
 
: <math>{d \over dx} \seccot x = -\tancsc^2 x = { -1 \secover \sin^2 x}</math>
 
: <math>{d \over dx} \cotcsc x = -\csc^2 x = { -1 \over \sin^2cot x}</math>
 
:<math>{d \over dx} \csc x = -\csc x \cot x</math>
| valign="top" |
: <math>{d \over dx} \mbox{bgsin} x = { 1 \over \sqrt{1 - x^2}}</math>
 
: <math>{d \over dx} \mbox{bgcos} x = {-1 \over \sqrt{1 - x^2}}</math>
 
: <math>{d \over dx} \mbox{bgtan} x = { 1 \over 1 + x^2}</math>
 
: <math>{d \over dx} \mbox{bgsec} x = { 1 \over |x|\sqrt{x^2 - 1}}</math>
 
: <math>{d \over dx} \mbox{bgcot} x = {-1 \over 1 + x^2}</math>
 
: <math>{d \over dx} \mbox{bgcsc} x = {-1 \over |x|\sqrt{x^2 - 1}}</math>
|}
 
== Afgeleides van [[hiperboliese funksie|hiperboliese funksies]]s ==
{|
|-
| valign="top" |
: <math>{d \over dx} \sinh x = \cosh x = \frac{e^x + e^{-x}}{2}</math>
 
: <math>{d \over dx} \cosh x = \sinh x = \frac{e^x - e^{-x}}{2}</math>
 
: <math>{d \over dx} \tanh x = \operatorname{sech}^2\,x</math>
 
: <math>{d \over dx}\,\operatorname{sech}\,x = - \tanh x\,\operatorname{sech}\,x</math>
 
: <math>{d \over dx}\,\operatorname{coth}\,x = -\,\operatorname{csch}^2\,x</math>
 
: <math>{d \over dx}\,\operatorname{csch}\,x = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x</math>
| valign="top" |
: <math>{d \over dx}\,\mbox{sinh} ^{-1} \,x = { 1 \over \sqrt{x^2 + 1}}</math>
 
: <math>{d \over dx}\,\mbox{cosh} ^{-1} \,x = { 1 \over \sqrt{x^2 - 1}}</math>
 
: <math>{d \over dx}\,\mbox{tanh} ^{-1} \,x = { 1 \over 1 - x^2}</math>
 
: <math>{d \over dx}\,\mbox{sech} ^{-1} \,x = { -1 \over x\sqrt{1 - x^2}}</math>
 
: <math>{d \over dx}\,\mbox{coth} ^{-1} \,x = { 1 \over 1 - x^2}</math>
 
: <math>{d \over dx}\,\mbox{csch} ^{-1} \,x = {-1 \over |x|\sqrt{1 + x^2}}</math>
 
|}
 
== Afgeleides van [[inverse funksie|inverse funksies]]s ==
: <math>{d \over dx} (f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}</math>
 
== Verwysings ==
# Stewart, J. (2003). ''Single Variable Calculus''. (5th ed.). Belmont, USA: Thomson Learning.
# Groenewald, G.J., Hitge, M. (2005). ''Analise II Studiegids vir WISK121A''. Potchefstroom: Noordwes-Universiteit.
# Jordan, D.W., Smith, P. (2002). ''Mathematical techniques: An introduction for the engineering, physical and mathematical sciences''. USA: Oxford University Press.
 
== Aantekeninge ==
* ''Sien ook [[Lys van integrale]]
 
* ''Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel [http[://en.wikipedia.org/wiki/Table_of_derivatives:Table of derivatives|Table of derivatives]]
 
[[Kategorie:Lyste|Afgeleides]]
 
[[es:Tabla de derivadas]]
[[pl:Pochodna funkcji#Pochodne_funkcji_elementarnychPochodne funkcji elementarnych]]