Operacion kryesor në njehsimin diferencial është gjetja e derivatit të një funksioni. Në këtë tabelë do të japim listën e derivateve të shumë funksioneve elementare. Në vazhdim, f dhe g janë funksione të derivueshme reale, dhe c është numër real.
Rregullat e përgjithshme të diferencimit(derivimit)
Redakto
Lineariteti
(
c
f
)
′
=
c
f
′
{\displaystyle \left({cf}\right)'=cf'}
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle \left({f+g}\right)'=f'+g'}
Rregulli i prodhimit
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle \left({fg}\right)'=f'g+fg'}
Derivati i funksionit reciprok
(
1
f
)
′
=
−
f
′
f
2
,
f
≠
0
{\displaystyle \left({\frac {1}{f}}\right)'={\frac {-f'}{f^{2}}},\qquad f\neq 0}
Derivati i herësit
(
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}
Derivati i funksionit të përbërë
(
f
∘
g
)
′
=
(
f
′
∘
g
)
g
′
{\displaystyle (f\circ g)'=(f'\circ g)g'}
Derivati i funksionit inverz
(
f
−
1
)
′
=
1
f
′
∘
f
−
1
{\displaystyle (f^{-1})'={\frac {1}{f'\circ f^{-1}}}}
Derivatet e funksioneve të thjeshta
Redakto
c
′
=
0
{\displaystyle c'=0\,}
x
′
=
1
{\displaystyle x'=1\,}
(
c
x
)
′
=
c
{\displaystyle (cx)'=c\,}
|
x
|
′
=
x
|
x
|
=
sgn
x
,
x
≠
0
{\displaystyle |x|'={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0}
(
x
c
)
′
=
c
x
c
−
1
{\displaystyle (x^{c})'=cx^{c-1}}
(
1
x
)
′
=
(
x
−
1
)
′
=
−
x
−
2
=
−
1
x
2
{\displaystyle \left({1 \over x}\right)'=\left(x^{-1}\right)'=-x^{-2}=-{1 \over x^{2}}}
(
1
x
c
)
′
=
(
x
−
c
)
′
=
−
c
x
−
c
−
1
=
−
c
x
c
+
1
{\displaystyle \left({1 \over x^{c}}\right)'=\left(x^{-c}\right)'=-cx^{-c-1}=-{c \over x^{c+1}}}
(
x
)
′
=
(
x
1
2
)
′
=
1
2
x
−
1
2
=
1
2
x
,
x
>
0
{\displaystyle \left({\sqrt {x}}\right)'=\left(x^{1 \over 2}\right)'={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}},\qquad x>0}
Derivatet e funksioneve eksponenciale dhe logaritmike
Redakto
Derivatet e funksioneve trigonometrike
Redakto
(
sin
x
)
′
=
cos
x
{\displaystyle (\sin x)'=\cos x\,}
(
arcsin
x
)
′
=
1
1
−
x
2
{\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}
(
cos
x
)
′
=
−
sin
x
{\displaystyle (\cos x)'=-\sin x\,}
(
arccos
x
)
′
=
−
1
1
−
x
2
{\displaystyle (\arccos x)'={-1 \over {\sqrt {1-x^{2}}}}\,}
(
tan
x
)
′
=
sec
2
x
=
1
cos
2
x
{\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}\,}
(
arctan
x
)
′
=
1
1
+
x
2
{\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}
(
sec
x
)
′
=
sec
x
tan
x
{\displaystyle (\sec x)'=\sec x\tan x\,}
(
arcsec
x
)
′
=
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
csc
x
)
′
=
−
csc
x
cot
x
{\displaystyle (\csc x)'=-\csc x\cot x\,}
(
arccsc
x
)
′
=
−
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arccsc} x)'={-1 \over |x|{\sqrt {x^{2}-1}}}\,}
(
cot
x
)
′
=
−
csc
2
x
=
−
1
sin
2
x
{\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}\,}
(
arccot
x
)
′
=
−
1
1
+
x
2
{\displaystyle (\operatorname {arccot} x)'={-1 \over 1+x^{2}}\,}
Derivatet e funksioneve hiperbolike
Redakto
(
sinh
x
)
′
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
(
arsinh
x
)
′
=
1
x
2
+
1
{\displaystyle (\operatorname {arsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}
(
cosh
x
)
′
=
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}
(
arcosh
x
)
′
=
1
x
2
−
1
{\displaystyle (\operatorname {arcosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}}
(
tanh
x
)
′
=
sech
2
x
{\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}
(
artanh
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {artanh} \,x)'={1 \over 1-x^{2}}}
(
sech
x
)
′
=
−
tanh
x
sech
x
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}
(
arsech
x
)
′
=
−
1
x
1
−
x
2
{\displaystyle (\operatorname {arsech} \,x)'={-1 \over x{\sqrt {1-x^{2}}}}}
(
csch
x
)
′
=
−
coth
x
csch
x
{\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}
(
arcsch
x
)
′
=
−
1
x
1
+
x
2
{\displaystyle (\operatorname {arcsch} \,x)'={-1 \over x{\sqrt {1+x^{2}}}}}
(
coth
x
)
′
=
−
csch
2
x
{\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}
(
arcoth
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {arcoth} \,x)'={1 \over 1-x^{2}}}
