Lys van afgeleides: Verskil tussen weergawes
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Saam met [[integrasie]] vorm [[differensiasie]] die hoofbewerkings van [[calculus]]. In die onderstaande lys is ''f'' en ''g'' differensieerbare [[funksie |
Saam met [[integrasie]] vorm [[differensiasie]] die hoofbewerkings van [[calculus]]. In die onderstaande lys is ''f'' en ''g'' differensieerbare [[funksie]]s van die [[reële]] [[getal]] ''s''. ''c'' is ook 'n reële getal. |
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Hierdie '''lys van afgeleides''' is voldoende om enige elementêre funksie te differensieer. |
Hierdie '''lys van afgeleides''' is voldoende om enige elementêre funksie te differensieer. |
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==Algemene reëls by die afleiding van funksies== |
== Algemene reëls by die afleiding van funksies == |
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:<math>\left({cf}\right)' = cf'</math> |
: <math>\left({cf}\right)' = cf'</math> |
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:<math>\left({f + g}\right)' = f' + g'</math> |
: <math>\left({f + g}\right)' = f' + g'</math> |
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;[[Produkreël]] |
; [[Produkreël]] |
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:<math>\left({fg}\right)' = f'g + fg'</math> |
: <math>\left({fg}\right)' = f'g + fg'</math> |
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;[[Kwosiëntreël]] |
; [[Kwosiëntreël]] |
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:<math>\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0</math> |
: <math>\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0</math> |
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;[[Kettingreël]] |
; [[Kettingreël]] |
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:<math>(f \circ g)' = (f' \circ g)g'</math> |
: <math>(f \circ g)' = (f' \circ g)g'</math> |
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==Afgeleides van eenvoudige funksies== |
== Afgeleides van eenvoudige funksies == |
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:<math>{d \over dx} c = 0</math> |
: <math>{d \over dx} c = 0</math> |
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:<math>{d \over dx} x = 1</math> |
: <math>{d \over dx} x = 1</math> |
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:<math>{d \over dx} cx = c</math> |
: <math>{d \over dx} cx = c</math> |
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:<math>{d \over dx} |x| = {|x| \over x} = \sgn x,\qquad x \ne 0</math> |
: <math>{d \over dx} |x| = {|x| \over x} = \sgn x,\qquad x \ne 0</math> |
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:<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} x^c = cx^{c-1} \qquad \mbox{met beide } x^c \mbox{ en } cx^{c-1} \mbox { gedefinieer}</math> |
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:<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}\right) = {d \over dx} \left(x^{-1}\right) = -x^{-2} = -{1 \over x^2}</math> |
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:<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} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -{c \over x^{c+1}}</math> |
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:<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> |
: <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> |
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==Afgeleides van [[eksponensiaalfunksies]] en [[logaritmes]]== |
== Afgeleides van [[eksponensiaalfunksies]] en [[logaritmes]] == |
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:<math>{d \over dx} c^x = {c^x \ln c },\qquad c > 0</math> |
: <math>{d \over dx} c^x = {c^x \ln c },\qquad c > 0</math> |
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:<math>{d \over dx} e^x = e^x</math> |
: <math>{d \over dx} e^x = e^x</math> |
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:<math>{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1</math> |
: <math>{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1</math> |
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:<math>{d \over dx} \ln x = {1 \over x},\qquad x > 0</math> |
: <math>{d \over dx} \ln x = {1 \over x},\qquad x > 0</math> |
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:<math>{d \over dx} \ln |x| = {1 \over x}</math> |
: <math>{d \over dx} \ln |x| = {1 \over x}</math> |
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:<math>{d \over dx} x^x = x^x(1+\ln x)</math> |
: <math>{d \over dx} x^x = x^x(1+\ln x)</math> |
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==Afgeleides van [[trigonometrie |
== Afgeleides van [[trigonometrie]]se funksies == |
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:<math>{d \over dx} \sin x = \cos x</math> |
: <math>{d \over dx} \sin x = \cos x</math> |
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:<math>{d \over dx} \cos x = -\sin x</math> |
: <math>{d \over dx} \cos x = -\sin x</math> |
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:<math>{d \over dx} \tan x = \sec^2 x = { 1 \over \cos^2 x}</math> |
: <math>{d \over dx} \tan x = \sec^2 x = { 1 \over \cos^2 x}</math> |
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⚫ | |||
:<math>{d \over dx} \ |
: <math>{d \over dx} \cot x = -\csc^2 x = { -1 \over \sin^2 x}</math> |
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:<math>{d \over dx} \ |
: <math>{d \over dx} \csc x = -\csc x \cot x</math> |
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:<math>{d \over dx} \mbox{bgsin} x = { 1 \over \sqrt{1 - x^2}}</math> |
: <math>{d \over dx} \mbox{bgsin} x = { 1 \over \sqrt{1 - x^2}}</math> |
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:<math>{d \over dx} \mbox{bgcos} x = {-1 \over \sqrt{1 - x^2}}</math> |
: <math>{d \over dx} \mbox{bgcos} x = {-1 \over \sqrt{1 - x^2}}</math> |
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:<math>{d \over dx} \mbox{bgtan} x = { 1 \over 1 + x^2}</math> |
: <math>{d \over dx} \mbox{bgtan} x = { 1 \over 1 + x^2}</math> |
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:<math>{d \over dx} \mbox{bgsec} x = { 1 \over |x|\sqrt{x^2 - 1}}</math> |
: <math>{d \over dx} \mbox{bgsec} x = { 1 \over |x|\sqrt{x^2 - 1}}</math> |
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:<math>{d \over dx} \mbox{bgcot} x = {-1 \over 1 + x^2}</math> |
: <math>{d \over dx} \mbox{bgcot} x = {-1 \over 1 + x^2}</math> |
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:<math>{d \over dx} \mbox{bgcsc} x = {-1 \over |x|\sqrt{x^2 - 1}}</math> |
: <math>{d \over dx} \mbox{bgcsc} x = {-1 \over |x|\sqrt{x^2 - 1}}</math> |
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== Afgeleides van [[hiperboliese funksie |
== Afgeleides van [[hiperboliese funksie]]s == |
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:<math>{d \over dx} \sinh x = \cosh x = \frac{e^x + e^{-x}}{2}</math> |
: <math>{d \over dx} \sinh x = \cosh x = \frac{e^x + e^{-x}}{2}</math> |
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:<math>{d \over dx} \cosh x = \sinh x = \frac{e^x - e^{-x}}{2}</math> |
: <math>{d \over dx} \cosh x = \sinh x = \frac{e^x - e^{-x}}{2}</math> |
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:<math>{d \over dx} \tanh x = \operatorname{sech}^2\,x</math> |
: <math>{d \over dx} \tanh x = \operatorname{sech}^2\,x</math> |
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:<math>{d \over dx}\,\operatorname{sech}\,x = - \tanh x\,\operatorname{sech}\,x</math> |
: <math>{d \over dx}\,\operatorname{sech}\,x = - \tanh x\,\operatorname{sech}\,x</math> |
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:<math>{d \over dx}\,\operatorname{coth}\,x = -\,\operatorname{csch}^2\,x</math> |
: <math>{d \over dx}\,\operatorname{coth}\,x = -\,\operatorname{csch}^2\,x</math> |
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:<math>{d \over dx}\,\operatorname{csch}\,x = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x</math> |
: <math>{d \over dx}\,\operatorname{csch}\,x = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x</math> |
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:<math>{d \over dx}\,\mbox{sinh} ^{-1} \,x = { 1 \over \sqrt{x^2 + 1}}</math> |
: <math>{d \over dx}\,\mbox{sinh} ^{-1} \,x = { 1 \over \sqrt{x^2 + 1}}</math> |
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:<math>{d \over dx}\,\mbox{cosh} ^{-1} \,x = { 1 \over \sqrt{x^2 - 1}}</math> |
: <math>{d \over dx}\,\mbox{cosh} ^{-1} \,x = { 1 \over \sqrt{x^2 - 1}}</math> |
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:<math>{d \over dx}\,\mbox{tanh} ^{-1} \,x = { 1 \over 1 - x^2}</math> |
: <math>{d \over dx}\,\mbox{tanh} ^{-1} \,x = { 1 \over 1 - x^2}</math> |
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:<math>{d \over dx}\,\mbox{sech} ^{-1} \,x = { -1 \over x\sqrt{1 - x^2}}</math> |
: <math>{d \over dx}\,\mbox{sech} ^{-1} \,x = { -1 \over x\sqrt{1 - x^2}}</math> |
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:<math>{d \over dx}\,\mbox{coth} ^{-1} \,x = { 1 \over 1 - x^2}</math> |
: <math>{d \over dx}\,\mbox{coth} ^{-1} \,x = { 1 \over 1 - x^2}</math> |
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:<math>{d \over dx}\,\mbox{csch} ^{-1} \,x = {-1 \over |x|\sqrt{1 + x^2}}</math> |
: <math>{d \over dx}\,\mbox{csch} ^{-1} \,x = {-1 \over |x|\sqrt{1 + x^2}}</math> |
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== Afgeleides van [[inverse funksie |
== Afgeleides van [[inverse funksie]]s == |
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:<math>{d \over dx} (f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}</math> |
: <math>{d \over dx} (f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}</math> |
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==Verwysings== |
== Verwysings == |
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#Stewart, J. (2003). ''Single Variable Calculus''. (5th ed.). Belmont, USA: Thomson Learning. |
# Stewart, J. (2003). ''Single Variable Calculus''. (5th ed.). Belmont, USA: Thomson Learning. |
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#Groenewald, G.J., Hitge, M. (2005). ''Analise II Studiegids vir WISK121A''. Potchefstroom: Noordwes-Universiteit. |
# Groenewald, G.J., Hitge, M. (2005). ''Analise II Studiegids vir WISK121A''. Potchefstroom: Noordwes-Universiteit. |
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#Jordan, D.W., Smith, P. (2002). ''Mathematical techniques: An introduction for the engineering, physical and mathematical sciences''. USA: Oxford University Press. |
# Jordan, D.W., Smith, P. (2002). ''Mathematical techniques: An introduction for the engineering, physical and mathematical sciences''. USA: Oxford University Press. |
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==Aantekeninge== |
== Aantekeninge == |
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*''Sien ook [[Lys van integrale]] |
* ''Sien ook [[Lys van integrale]] |
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*''Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel [ |
* ''Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel [[:en:Table of derivatives|Table of derivatives]] |
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[[Kategorie:Lyste|Afgeleides]] |
[[Kategorie:Lyste|Afgeleides]] |
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Lyn 122: | Lyn 121: | ||
[[es:Tabla de derivadas]] |
[[es:Tabla de derivadas]] |
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[[pl:Pochodna funkcji# |
[[pl:Pochodna funkcji#Pochodne funkcji elementarnych]] |
Wysiging soos op 12:36, 7 November 2015
Saam met integrasie vorm differensiasie die hoofbewerkings van calculus. In die onderstaande lys is f en g differensieerbare funksies 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
Afgeleides van eenvoudige funksies
Afgeleides van eksponensiaalfunksies en logaritmes
Afgeleides van trigonometriese funksies
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Afgeleides van hiperboliese funksies
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Afgeleides van inverse funksies
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 Table of derivatives