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In matematica , le regole di derivazione e le derivate fondamentali sono regole studiate per evitare di dover calcolare ogni volta il limite del rapporto incrementale di funzioni , e utilizzate al fine di facilitare la derivazione di funzioni di maggiore complessità.
Siano
f
(
x
)
{\displaystyle f(x)}
g
(
x
)
{\displaystyle g(x)}
x
{\displaystyle x}
D
{\displaystyle \mathrm {D} }
x
{\displaystyle x}
D
[
f
(
x
)
]
=
f
′
(
x
)
,
D
[
g
(
x
)
]
=
g
′
(
x
)
.
{\displaystyle \mathrm {D} [f(x)]=f'(x),\qquad \mathrm {D} [g(x)]=g'(x).}
D
[
α
f
(
x
)
+
β
g
(
x
)
]
=
α
f
′
(
x
)
+
β
g
′
(
x
)
,
α
,
β
∈
R
.
{\displaystyle \mathrm {D} [\alpha f(x)+\beta g(x)]=\alpha f'(x)+\beta g'(x),\qquad \alpha ,\beta \in \mathbb {R} .}
D
[
f
(
x
)
⋅
g
(
x
)
]
=
f
′
(
x
)
⋅
g
(
x
)
+
f
(
x
)
⋅
g
′
(
x
)
.
{\displaystyle \mathrm {D} [{f(x)\cdot g(x)}]=f'(x)\cdot g(x)+f(x)\cdot g'(x).}
D
[
f
(
x
)
g
(
x
)
]
=
f
′
(
x
)
⋅
g
(
x
)
−
f
(
x
)
⋅
g
′
(
x
)
g
(
x
)
2
.
{\displaystyle \mathrm {D} \left[{f(x) \over g(x)}\right]={f'(x)\cdot g(x)-f(x)\cdot g'(x) \over g(x)^{2}}.}
D
[
1
f
(
x
)
]
=
−
f
′
(
x
)
f
(
x
)
2
.
{\displaystyle \mathrm {D} \left[{1 \over f(x)}\right]=-{f'(x) \over f(x)^{2}}.}
D
[
f
−
1
(
x
)
]
=
1
f
′
(
f
−
1
(
x
)
)
.
{\displaystyle \mathrm {D} [f^{-1}(x)]={1 \over f'(f^{-1}(x))}.}
D
[
f
(
g
(
x
)
)
]
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
.
{\displaystyle \mathrm {D} \left[f\left(g(x)\right)\right]=f'\left(g(x)\right)\cdot g'(x).}
D
[
f
(
x
)
g
(
x
)
]
=
f
(
x
)
g
(
x
)
[
g
′
(
x
)
ln
(
f
(
x
)
)
+
g
(
x
)
f
′
(
x
)
f
(
x
)
]
.
{\displaystyle \mathrm {D} \left[f(x)^{g(x)}\right]=f(x)^{g(x)}\left[g'(x)\ln(f(x))+{\frac {g(x)f'(x)}{f(x)}}\right].}
Ognuna di queste funzioni, se non altrimenti specificato, è derivabile in tutto il suo campo di esistenza .
D
(
a
)
=
0
,
a
costante
.
{\displaystyle \mathrm {D} (a)=0,\qquad a{\text{ costante}}.}
D
(
x
)
=
1.
{\displaystyle \mathrm {D} (x)=1.}
D
(
a
x
)
=
a
,
a
costante
.
{\displaystyle \mathrm {D} (ax)=a,\qquad a{\text{ costante}}.}
D
(
x
2
)
=
2
x
.
{\displaystyle \mathrm {D} (x^{2})=2x.}
D
(
x
3
)
=
3
x
2
.
{\displaystyle \mathrm {D} (x^{3})=3x^{2}.}
Più in generale si ha:
D
(
x
n
)
=
n
x
n
−
1
,
con
n
∈
N
.
{\displaystyle \mathrm {D} (x^{n})=nx^{n-1},\qquad {\text{con }}n\in \mathbb {N} .}
Da quest'ultima relazione segue che se
f
(
x
)
{\displaystyle f(x)}
n
{\displaystyle n}
D
(
f
(
x
)
)
{\displaystyle D\left(f(x)\right)}
n
−
1
{\displaystyle n-1}
D
(
x
α
)
=
α
x
α
−
1
,
con
α
∈
R
.
{\displaystyle \mathrm {D} (x^{\alpha })=\alpha x^{\alpha -1},\qquad {\text{con }}\alpha \in \mathbb {R} .}
D
(
x
2
)
=
1
2
x
2
.
{\displaystyle \mathrm {D} ({\sqrt[{2}]{x}})={\frac {1}{2{\sqrt[{2}]{x}}}}.}
D
(
x
m
n
)
=
m
n
x
m
−
n
n
,
se
x
>
0.
{\displaystyle \mathrm {D} ({\sqrt[{n}]{x^{m}}})={{\frac {m}{n}}{\sqrt[{n}]{x^{m-n}}}},\qquad {\text{se }}x>0.}
D
(
|
x
|
)
=
|
x
|
x
=
x
|
x
|
.
{\displaystyle \mathrm {D} (|x|)={\dfrac {|x|}{x}}={\dfrac {x}{|x|}}.}
D
(
log
b
x
)
=
log
b
e
x
=
1
x
ln
b
.
{\displaystyle \mathrm {D} (\log _{b}x)={\frac {\log _{b}\mathrm {e} }{x}}={\frac {1}{x\ln b}}.}
D
(
ln
x
)
=
1
x
.
{\displaystyle \mathrm {D} (\ln x)={\frac {1}{x}}.}
D
(
e
x
)
=
e
x
.
{\displaystyle \mathrm {D} (e^{x})=\mathrm {e} ^{x}.}
D
(
a
x
)
=
a
x
ln
a
.
{\displaystyle \mathrm {D} (a^{x})=a^{x}\ln a.}
D
(
x
x
)
=
x
x
(
1
+
ln
x
)
.
{\displaystyle \mathrm {D} (x^{x})=x^{x}(1+\ln x).}
D
(
sin
x
)
=
cos
x
.
{\displaystyle \mathrm {D} (\sin x)=\cos x.}
D
(
cos
x
)
=
−
sin
x
.
{\displaystyle \mathrm {D} (\cos x)=-\sin x.}
D
(
tan
x
)
=
1
+
tan
2
x
=
1
cos
2
x
.
{\displaystyle \mathrm {D} (\tan x)=1+\tan ^{2}x={1 \over \cos ^{2}x}.}
D
(
cot
x
)
=
−
(
1
+
cot
2
x
)
=
−
1
sin
2
x
.
{\displaystyle \mathrm {D} (\cot x)=-(1+\cot ^{2}x)=-{\frac {1}{\sin ^{2}x}}.}
D
(
sec
x
)
=
tan
x
sec
x
.
{\displaystyle \mathrm {D} (\sec x)=\tan x\sec x.}
D
(
csc
x
)
=
−
cot
x
csc
x
.
{\displaystyle \mathrm {D} (\csc x)=-\cot x\csc x.}
D
(
arcsin
x
)
=
1
1
−
x
2
.
{\displaystyle \mathrm {D} (\arcsin x)={\frac {1}{\sqrt {1-x^{2}}}}.}
D
(
arccos
x
)
=
−
1
1
−
x
2
.
{\displaystyle \mathrm {D} (\arccos x)=-{\frac {1}{\sqrt {1-x^{2}}}}.}
D
(
arctan
x
)
=
1
1
+
x
2
.
{\displaystyle \mathrm {D} (\arctan x)={\frac {1}{1+x^{2}}}.}
D
(
arccot
x
)
=
−
1
1
+
x
2
.
{\displaystyle \mathrm {D} (\operatorname {arccot} x)={-1 \over 1+x^{2}}.}
D
(
arcsec
x
)
=
1
|
x
|
x
2
−
1
.
{\displaystyle \mathrm {D} (\operatorname {arcsec} x)={1 \over |x|{\sqrt {x^{2}-1}}}.}
D
(
arccsc
x
)
=
−
1
|
x
|
x
2
−
1
.
{\displaystyle \mathrm {D} (\operatorname {arccsc} x)={-1 \over |x|{\sqrt {x^{2}-1}}}.}
D
(
sinh
x
)
=
cosh
x
.
{\displaystyle \mathrm {D} (\sinh x)=\cosh x.}
D
(
cosh
x
)
=
sinh
x
.
{\displaystyle \mathrm {D} (\cosh x)=\sinh x.}
D
(
tanh
x
)
=
1
−
tanh
2
x
=
1
cosh
2
x
.
{\displaystyle \mathrm {D} (\tanh x)=1-\tanh ^{2}x={1 \over \cosh ^{2}x}.}
D
(
coth
x
)
=
−
csch
2
x
.
{\displaystyle \mathrm {D} ({\mbox{coth}}\,x)=-{\mbox{csch}}^{2}\,x.}
D
(
sech
x
)
=
−
tanh
x
sech
x
.
{\displaystyle \mathrm {D} ({\mbox{sech}}\,x)=-\tanh x\;{\mbox{sech}}\,x.}
D
(
csch
x
)
=
−
coth
x
csch
x
.
{\displaystyle \mathrm {D} ({\mbox{csch}}\,x)=-{\mbox{coth}}\,x\;{\mbox{csch}}\,x.}
D
(
settsinh
x
)
=
1
x
2
+
1
.
{\displaystyle \mathrm {D} ({\mbox{settsinh}}\,x)={1 \over {\sqrt {x^{2}+1}}}.}
D
(
settanh
x
)
=
1
1
−
x
2
.
{\displaystyle \mathrm {D} ({\mbox{settanh}}\,x)={1 \over 1-x^{2}}.}
D
(
settcoth
x
)
=
1
1
−
x
2
.
{\displaystyle \mathrm {D} ({\mbox{settcoth}}\,x)={1 \over 1-x^{2}}.}
D
(
settsech
x
)
=
−
1
x
1
−
x
2
.
{\displaystyle \mathrm {D} ({\mbox{settsech}}\,x)={-1 \over x{\sqrt {1-x^{2}}}}.}
D
(
settcsch
x
)
=
−
1
|
x
|
1
+
x
2
.
{\displaystyle \mathrm {D} ({\mbox{settcsch}}\,x)={-1 \over |x|{\sqrt {1+x^{2}}}}.}
D
(
|
f
(
x
)
|
)
=
f
′
(
x
)
f
(
x
)
|
f
(
x
)
|
=
f
′
(
x
)
|
f
(
x
)
|
f
(
x
)
.
{\displaystyle \mathrm {D} (|f(x)|)=f'(x){\dfrac {f(x)}{|f(x)|}}=f'(x){\dfrac {|f(x)|}{f(x)}}.}
D
(
[
f
(
x
)
]
n
)
=
n
⋅
f
(
x
)
n
−
1
⋅
f
′
(
x
)
.
{\displaystyle \mathrm {D} ([f(x)]^{n})=n\cdot f(x)^{n-1}\cdot f'(x).}
D
(
ln
f
(
x
)
)
=
f
′
(
x
)
f
(
x
)
.
{\displaystyle \mathrm {D} (\ln f(x))={f'(x) \over f(x)}.}
D
(
ln
|
f
(
x
)
|
)
=
f
′
(
x
)
f
(
x
)
.
{\displaystyle \mathrm {D} (\ln |f(x)|)={f'(x) \over f(x)}.}
D
(
e
f
(
x
)
)
=
e
f
(
x
)
⋅
f
′
(
x
)
.
{\displaystyle \mathrm {D} (\mathrm {e} ^{f(x)})=\mathrm {e} ^{f(x)}\cdot f'(x).}
D
(
a
f
(
x
)
)
=
a
f
(
x
)
⋅
f
′
(
x
)
⋅
ln
a
.
{\displaystyle \mathrm {D} (a^{f(x)})=a^{f(x)}\cdot f'(x)\cdot \ln a.}
D
(
sin
f
(
x
)
)
=
cos
f
(
x
)
⋅
f
′
(
x
)
.
{\displaystyle \mathrm {D} (\sin f(x))=\cos f(x)\cdot f'(x).}
D
(
cos
f
(
x
)
)
=
−
sin
f
(
x
)
⋅
f
′
(
x
)
.
{\displaystyle \mathrm {D} (\cos f(x))=-\sin f(x)\cdot f'(x).}
D
(
tan
f
(
x
)
)
=
f
′
(
x
)
cos
2
f
(
x
)
.
{\displaystyle \mathrm {D} (\tan f(x))={f'(x) \over \cos ^{2}f(x)}.}
D
(
arcsin
f
(
x
)
)
=
f
′
(
x
)
1
−
[
f
(
x
)
]
2
.
{\displaystyle D(\arcsin f(x))={f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.}
D
(
arccos
f
(
x
)
)
=
−
f
′
(
x
)
1
−
[
f
(
x
)
]
2
.
{\displaystyle D(\arccos f(x))={-f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.}
D
(
arctan
f
(
x
)
)
=
f
′
(
x
)
1
+
[
f
(
x
)
]
2
.
{\displaystyle D(\arctan f(x))={f'(x) \over 1+[f(x)]^{2}}.}
D
(
f
(
x
)
g
(
x
)
)
=
f
(
x
)
g
(
x
)
⋅
[
g
′
(
x
)
⋅
ln
f
(
x
)
+
g
(
x
)
⋅
f
′
(
x
)
f
(
x
)
]
.
{\displaystyle D(f(x)^{g(x)})=f(x)^{g(x)}\cdot \left[g'(x)\cdot \ln f(x)+g(x)\cdot {f'(x) \over f(x)}\right].}
Dimostrazione
f
(
x
)
g
(
x
)
=
e
ln
f
(
x
)
g
(
x
)
=
e
g
(
x
)
⋅
ln
f
(
x
)
{\displaystyle {f(x)^{g(x)}}={e^{{\ln }{f(x)^{g(x)}}}}={e^{g(x)\cdot {{\ln }f(x)}}}}
D
(
e
f
(
x
)
)
{\displaystyle D({e^{f(x)}})}
D
(
x
f
(
x
)
)
=
x
f
(
x
)
⋅
[
f
′
(
x
)
⋅
ln
x
+
f
(
x
)
x
]
.
{\displaystyle D(x^{f(x)})=x^{f(x)}\cdot \left[f'(x)\cdot \ln x+{f(x) \over x}\right].}
![{\displaystyle \mathrm {D} [f(x)]=f'(x),\qquad \mathrm {D} [g(x)]=g'(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3dbbee591cfc03840aee7b3156fbb214dc15c9)
![{\displaystyle \mathrm {D} [\alpha f(x)+\beta g(x)]=\alpha f'(x)+\beta g'(x),\qquad \alpha ,\beta \in \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a6500ce06fa522a6c5d5a6c9e3b8a2579055ad)
![{\displaystyle \mathrm {D} [{f(x)\cdot g(x)}]=f'(x)\cdot g(x)+f(x)\cdot g'(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74398f82204979f9097b8f0c739a7fb72bf9dd4f)
![{\displaystyle \mathrm {D} \left[{f(x) \over g(x)}\right]={f'(x)\cdot g(x)-f(x)\cdot g'(x) \over g(x)^{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5ab95aa8459c3cd59bb25cd94dcb9be7beca56)
![{\displaystyle \mathrm {D} \left[{1 \over f(x)}\right]=-{f'(x) \over f(x)^{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab55df5e27c8a878d420d8bb9b5297a7ac46713)
![{\displaystyle \mathrm {D} [f^{-1}(x)]={1 \over f'(f^{-1}(x))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e06882f1975b6d8fa37dd57fcd36cd0cecc0c6f5)
![{\displaystyle \mathrm {D} \left[f\left(g(x)\right)\right]=f'\left(g(x)\right)\cdot g'(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2817599e9674d0fc0091e8c197a91595e3e4ff29)
![{\displaystyle \mathrm {D} \left[f(x)^{g(x)}\right]=f(x)^{g(x)}\left[g'(x)\ln(f(x))+{\frac {g(x)f'(x)}{f(x)}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5372f08f78cf6feb831e90307bb4f2eaed3fbb78)
![{\displaystyle \mathrm {D} (f(x))=\lim _{h\to 0}{{f(x+h)-f(x)} \over {h}}=\lim _{h\to 0}{\sum _{k=0}^{n}{a_{k}(x+h)^{k}}-\sum _{k=0}^{n}{a_{k}x^{k}} \over h}=\lim _{h\to 0}{\sum _{k=0}^{n}{a_{k}\left[(x+h)^{k}-x^{k}\right]} \over h}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/931e62d3198e1e2dd942926fe28f9290683410f7)
![{\displaystyle \mathrm {D} ({\sqrt[{2}]{x}})={\frac {1}{2{\sqrt[{2}]{x}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa51b242cf9d670df7ee31a03ac62f2e67ffc33d)
![{\displaystyle \mathrm {D} ({\sqrt[{n}]{x^{m}}})={{\frac {m}{n}}{\sqrt[{n}]{x^{m-n}}}},\qquad {\text{se }}x>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b422f7b60f37c57b34c696546e2de446b0c5b6)
![{\displaystyle \mathrm {D} ({\sqrt[{n}]{x^{m}}})=\mathrm {D} \left(x^{\frac {m}{n}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69188f3a2a8efe1f5c142dd7c96dd29839734009)
![{\displaystyle \mathrm {D} ({\sqrt[{n}]{x^{m}}})={\frac {m}{n}}x^{{\frac {m}{n}}-1}={\frac {m}{n}}x^{\frac {m-n}{n}}={\frac {m}{n}}{\sqrt[{n}]{x^{m-n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52afce82b7b278fd9f8250a5aa44a2cfc666a5f6)
![{\displaystyle \mathrm {D} ([f(x)]^{n})=n\cdot f(x)^{n-1}\cdot f'(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acda3dc305bb142e50d63a30ffe67dabde6813d1)
![{\displaystyle D(\arcsin f(x))={f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63fd316798cefbec6529fe48b94a637c4667bf29)
![{\displaystyle D(\arccos f(x))={-f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/394120c0b30ba4221a698d7dc4c6874ddcffdd5a)
![{\displaystyle D(\arctan f(x))={f'(x) \over 1+[f(x)]^{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9544bbb24e2189c0c44f3f8537c5b34dfc5084d)
![{\displaystyle D(f(x)^{g(x)})=f(x)^{g(x)}\cdot \left[g'(x)\cdot \ln f(x)+g(x)\cdot {f'(x) \over f(x)}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba6b22fbae869aef2d2c9dd3f02a9d8d83b832a)
![{\displaystyle D(x^{f(x)})=x^{f(x)}\cdot \left[f'(x)\cdot \ln x+{f(x) \over x}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db26b76af91ca76fc8acaf7ee36494cd2603270c)
