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Lyn 1:
Lyn 1:
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.
Saam met [[integrasie]] vorm [[differensiasie]] die hoofbewerkings van [[calculus|kalkulus ]]. In die onderstaande lys is ''f'' en ''g'' differensieerbare [[funksie]]s van die [[reële]] [[getal]] ''s''. ''c'' is ook 'n reële getal.
Hierdie '''lys van afgeleides''' is voldoende om enige elementêre funksie te differensieer.
Hierdie '''lys van afgeleides''' is voldoende om enige elementêre funksie te differensieer.
Wysiging soos op 23:36, 12 Desember 2015
Saam met integrasie vorm differensiasie die hoofbewerkings van kalkulus . 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
(
c
f
)
′
=
c
f
′
{\displaystyle \left({cf}\right)'=cf'}
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle \left({f+g}\right)'=f'+g'}
Produkreël
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle \left({fg}\right)'=f'g+fg'}
Kwosiëntreël
(
f
g
)
′
=
f
′
g
−
f
g
′
g
2
,
g
≠
0
{\displaystyle \left({f \over g}\right)'={f'g-fg' \over g^{2}},\qquad g\neq 0}
Kettingreël
(
f
∘
g
)
′
=
(
f
′
∘
g
)
g
′
{\displaystyle (f\circ g)'=(f'\circ g)g'}
Afgeleides van eenvoudige funksies
d
d
x
c
=
0
{\displaystyle {d \over dx}c=0}
d
d
x
x
=
1
{\displaystyle {d \over dx}x=1}
d
d
x
c
x
=
c
{\displaystyle {d \over dx}cx=c}
d
d
x
|
x
|
=
|
x
|
x
=
sgn
x
,
x
≠
0
{\displaystyle {d \over dx}|x|={|x| \over x}=\operatorname {sgn} x,\qquad x\neq 0}
d
d
x
x
c
=
c
x
c
−
1
met beide
x
c
en
c
x
c
−
1
gedefinieer
{\displaystyle {d \over dx}x^{c}=cx^{c-1}\qquad {\mbox{met beide }}x^{c}{\mbox{ en }}cx^{c-1}{\mbox{ gedefinieer}}}
d
d
x
(
1
x
)
=
d
d
x
(
x
−
1
)
=
−
x
−
2
=
−
1
x
2
{\displaystyle {d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2}}}
d
d
x
(
1
x
c
)
=
d
d
x
(
x
−
c
)
=
−
c
x
c
+
1
{\displaystyle {d \over dx}\left({1 \over x^{c}}\right)={d \over dx}\left(x^{-c}\right)=-{c \over x^{c+1}}}
d
d
x
x
=
d
d
x
x
1
2
=
1
2
x
−
1
2
=
1
2
x
,
x
>
0
{\displaystyle {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}
d
d
x
c
x
=
c
x
ln
c
,
c
>
0
{\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0}
d
d
x
e
x
=
e
x
{\displaystyle {d \over dx}e^{x}=e^{x}}
d
d
x
log
c
x
=
1
x
ln
c
,
c
>
0
,
c
≠
1
{\displaystyle {d \over dx}\log _{c}x={1 \over x\ln c},\qquad c>0,c\neq 1}
d
d
x
ln
x
=
1
x
,
x
>
0
{\displaystyle {d \over dx}\ln x={1 \over x},\qquad x>0}
d
d
x
ln
|
x
|
=
1
x
{\displaystyle {d \over dx}\ln |x|={1 \over x}}
d
d
x
x
x
=
x
x
(
1
+
ln
x
)
{\displaystyle {d \over dx}x^{x}=x^{x}(1+\ln x)}
d
d
x
sin
x
=
cos
x
{\displaystyle {d \over dx}\sin x=\cos x}
d
d
x
cos
x
=
−
sin
x
{\displaystyle {d \over dx}\cos x=-\sin x}
d
d
x
tan
x
=
sec
2
x
=
1
cos
2
x
{\displaystyle {d \over dx}\tan x=\sec ^{2}x={1 \over \cos ^{2}x}}
d
d
x
sec
x
=
tan
x
sec
x
{\displaystyle {d \over dx}\sec x=\tan x\sec x}
d
d
x
cot
x
=
−
csc
2
x
=
−
1
sin
2
x
{\displaystyle {d \over dx}\cot x=-\csc ^{2}x={-1 \over \sin ^{2}x}}
d
d
x
csc
x
=
−
csc
x
cot
x
{\displaystyle {d \over dx}\csc x=-\csc x\cot x}
d
d
x
bgsin
x
=
1
1
−
x
2
{\displaystyle {d \over dx}{\mbox{bgsin}}x={1 \over {\sqrt {1-x^{2}}}}}
d
d
x
bgcos
x
=
−
1
1
−
x
2
{\displaystyle {d \over dx}{\mbox{bgcos}}x={-1 \over {\sqrt {1-x^{2}}}}}
d
d
x
bgtan
x
=
1
1
+
x
2
{\displaystyle {d \over dx}{\mbox{bgtan}}x={1 \over 1+x^{2}}}
d
d
x
bgsec
x
=
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}{\mbox{bgsec}}x={1 \over |x|{\sqrt {x^{2}-1}}}}
d
d
x
bgcot
x
=
−
1
1
+
x
2
{\displaystyle {d \over dx}{\mbox{bgcot}}x={-1 \over 1+x^{2}}}
d
d
x
bgcsc
x
=
−
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}{\mbox{bgcsc}}x={-1 \over |x|{\sqrt {x^{2}-1}}}}
d
d
x
sinh
x
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle {d \over dx}\sinh x=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
d
d
x
cosh
x
=
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle {d \over dx}\cosh x=\sinh x={\frac {e^{x}-e^{-x}}{2}}}
d
d
x
tanh
x
=
sech
2
x
{\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}\,x}
d
d
x
sech
x
=
−
tanh
x
sech
x
{\displaystyle {d \over dx}\,\operatorname {sech} \,x=-\tanh x\,\operatorname {sech} \,x}
d
d
x
coth
x
=
−
csch
2
x
{\displaystyle {d \over dx}\,\operatorname {coth} \,x=-\,\operatorname {csch} ^{2}\,x}
d
d
x
csch
x
=
−
coth
x
csch
x
{\displaystyle {d \over dx}\,\operatorname {csch} \,x=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}
d
d
x
sinh
−
1
x
=
1
x
2
+
1
{\displaystyle {d \over dx}\,{\mbox{sinh}}^{-1}\,x={1 \over {\sqrt {x^{2}+1}}}}
d
d
x
cosh
−
1
x
=
1
x
2
−
1
{\displaystyle {d \over dx}\,{\mbox{cosh}}^{-1}\,x={1 \over {\sqrt {x^{2}-1}}}}
d
d
x
tanh
−
1
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\,{\mbox{tanh}}^{-1}\,x={1 \over 1-x^{2}}}
d
d
x
sech
−
1
x
=
−
1
x
1
−
x
2
{\displaystyle {d \over dx}\,{\mbox{sech}}^{-1}\,x={-1 \over x{\sqrt {1-x^{2}}}}}
d
d
x
coth
−
1
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\,{\mbox{coth}}^{-1}\,x={1 \over 1-x^{2}}}
d
d
x
csch
−
1
x
=
−
1
|
x
|
1
+
x
2
{\displaystyle {d \over dx}\,{\mbox{csch}}^{-1}\,x={-1 \over |x|{\sqrt {1+x^{2}}}}}
d
d
x
(
f
−
1
(
x
)
)
=
1
f
′
(
f
−
1
(
x
)
)
{\displaystyle {d \over dx}(f^{-1}(x))={\frac {1}{f'(f^{-1}(x))}}}
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.
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