Spisak diferencijacijskih identiteta
Primarna operacija u diferencijalnom kalkulusu je računanje derivacije. Ova tabela sadrži derivacije nekih osnovnih funkcija . U sljedećem tekstu, f i g su diferencijabilne funkciju u skupu realnih brojeva , a c je realan broj. Ove formule su dovoljne za izračunavanje derivacija bilo koje elementarne funkcije.
Opća pravila diferenciranja
Linearnost
(
c
f
)
′
=
c
f
′
{\displaystyle \left({cf}\right)'=cf'}
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle \left({f+g}\right)'=f'+g'}
(
f
−
g
)
′
=
f
′
−
g
′
{\displaystyle \left({f-g}\right)'=f'-g'}
Pravilo derivacije proizvoda
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle \left({fg}\right)'=f'g+fg'}
Pravilo derivacije količnika
(
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}
Pravilo derivacije funkcije sa potencijom
(
f
g
)
′
=
(
e
g
ln
f
)
′
=
f
g
(
f
′
g
f
+
g
′
ln
f
)
,
f
>
0
{\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\qquad f>0}
Pravilo derivacije složene funkcije
(
f
∘
g
)
′
=
(
f
′
∘
g
)
g
′
{\displaystyle (f\circ g)'=(f'\circ g)g'}
Pravilo derivacije logaritma
f
′
=
(
ln
f
)
′
f
,
f
>
0
{\displaystyle f'=(\ln f)'f,\qquad f>0}
Derivacije jednostavnih funkcija
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
gdje su i
x
c
i
c
x
c
−
1
definisane
{\displaystyle {d \over dx}x^{c}=cx^{c-1}\qquad {\mbox{gdje su i }x^{c}{\mbox{ i }cx^{c-1}{\mbox{ definisane}
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}
Derivacije eksponencijalnih i logaritamskih funkcija
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
arcsin
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2}
d
d
x
arccos
x
=
−
1
1
−
x
2
{\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}
d
d
x
arctan
x
=
1
1
+
x
2
{\displaystyle {d \over dx}\arctan x={1 \over 1+x^{2}
d
d
x
arcsec
x
=
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}
d
d
x
arccot
x
=
−
1
1
+
x
2
{\displaystyle {d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2}
d
d
x
arccsc
x
=
−
1
|
x
|
x
2
−
1
{\displaystyle {d \over dx}\operatorname {arccsc} 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
arcsinh
x
=
1
x
2
+
1
{\displaystyle {d \over dx}\,\operatorname {arcsinh} \,x={1 \over {\sqrt {x^{2}+1}
d
d
x
arccosh
x
=
1
x
2
−
1
{\displaystyle {d \over dx}\,\operatorname {arccosh} \,x={1 \over {\sqrt {x^{2}-1}
d
d
x
arctanh
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\,\operatorname {arctanh} \,x={1 \over 1-x^{2}
d
d
x
arcsech
x
=
−
1
x
1
−
x
2
{\displaystyle {d \over dx}\,\operatorname {arcsech} \,x={-1 \over x{\sqrt {1-x^{2}
d
d
x
arccoth
x
=
1
1
−
x
2
{\displaystyle {d \over dx}\,\operatorname {arccoth} \,x={1 \over 1-x^{2}
d
d
x
arccsch
x
=
−
1
|
x
|
1
+
x
2
{\displaystyle {d \over dx}\,\operatorname {arccsch} \,x={-1 \over |x|{\sqrt {1+x^{2}
Također pogledajte
The article is a derivative under the Creative Commons Attribution-ShareAlike License .
A link to the original article can be found here and attribution parties here
By using this site, you agree to the Terms of Use . Gpedia ® is a registered trademark of the Cyberajah Pty Ltd