الرئيسية
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مشتقات عادية
مشتقات
الدوال
الاعتيادية :
مجال الدالة
D
f
{\displaystyle D_{f}\,\!}
الدالة
f
(
x
)
{\displaystyle f(x)\,\!}
مجال المشتقة
D
f
′
{\displaystyle D_{f'}\,\!}
المشتقة
f
′
(
x
)
{\displaystyle f'(x)\,\!}
تعليق
R
{\displaystyle \mathbb {R} \,\!}
k
{\displaystyle k\,\!}
R
{\displaystyle \mathbb {R} \,\!}
0
{\displaystyle 0\,\!}
k
∈
R
{\displaystyle k\in \mathbb {R} }
R
{\displaystyle \mathbb {R} \,\!}
x
{\displaystyle x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
1
{\displaystyle 1\,\!}
حال
x
n
{\displaystyle x^{n}}
عند
n
=
1
{\displaystyle n=1}
R
{\displaystyle \mathbb {R} \,\!}
x
2
{\displaystyle x^{2}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
2
x
{\displaystyle 2x\,\!}
حالة
n
=
2
{\displaystyle n=2}
عند
x
n
{\displaystyle x^{n}}
R
+
{\displaystyle \mathbb {R} _{+}\,\!}
x
{\displaystyle {\sqrt {x}}\,\!}
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}\,\!}
1
2
x
{\displaystyle {\frac {1}{2{\sqrt {x}}}}\,\!}
حالة
x
α
{\displaystyle x^{\alpha }}
عند
α
=
1
/
2
{\displaystyle \alpha =1/2}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
1
x
{\displaystyle {\frac {1}{x}}\,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
−
1
x
2
{\displaystyle -{\frac {1}{x^{2}}}\,\!}
حالة
1
/
x
n
{\displaystyle 1/x^{n}}
عند
n
=
1
{\displaystyle n=1}
R
{\displaystyle \mathbb {R} \,\!}
x
n
{\displaystyle x^{n}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
n
x
n
−
1
{\displaystyle nx^{n-1}\,\!}
n
∈
N
{\displaystyle n\in \mathbb {N} \,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
1
x
n
{\displaystyle {\frac {1}{x^{n}}}\,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
−
n
x
n
+
1
{\displaystyle -{\frac {n}{x^{n+1}}}\,\!}
n
∈
N
{\displaystyle n\in \mathbb {N} \,\!}
R
+
{\displaystyle \mathbb {R} _{+}\,\!}
x
n
{\displaystyle {\sqrt[{n}]{x}}\,\!}
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}\,\!}
1
n
x
n
−
1
n
{\displaystyle {\frac {1}{n{\sqrt[{n}]{x^{n-1}}}}}\,\!}
n
∈
N
{\displaystyle n\in \mathbb {N} ~}
، حالة
x
α
{\displaystyle x^{\alpha }}
عند
α
=
1
/
n
{\displaystyle \alpha =1/n}
R
+
{\displaystyle \mathbb {R} _{+}\,\!}
x
α
{\displaystyle x^{\alpha }\,\!}
R
+
{\displaystyle \mathbb {R} _{+}\,\!}
α
x
α
−
1
{\displaystyle \alpha x^{\alpha -1}\,\!}
α
≥
1
{\displaystyle \alpha \geq 1\,\!}
R
+
{\displaystyle \mathbb {R} _{+}\,\!}
x
α
{\displaystyle x^{\alpha }\,\!}
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}\,\!}
α
x
α
−
1
{\displaystyle \alpha x^{\alpha -1}\,\!}
0
<
α
<
1
{\displaystyle 0<\alpha <1\,\!}
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}\,\!}
x
α
{\displaystyle x^{\alpha }\,\!}
R
+
∗
{\displaystyle \mathbb {R} _{+}^{*}\,\!}
α
x
α
−
1
{\displaystyle \alpha x^{\alpha -1}\,\!}
α
<
0
{\displaystyle \alpha <0\,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
ln
|
x
|
{\displaystyle \ln |x|\,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
1
x
{\displaystyle {\frac {1}{x}}\,\!}
حالة
log
a
x
{\displaystyle \log _{a}x}
عتد
a
=e
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
log
a
|
x
|
{\displaystyle \log _{a}|x|\,\!}
R
∗
{\displaystyle \mathbb {R} ^{*}\,\!}
1
x
ln
a
{\displaystyle {\frac {1}{x\ln a}}\,\!}
a
>
0
{\displaystyle a>0\,\!}
R
{\displaystyle \mathbb {R} \,\!}
e
x
{\displaystyle e^{x}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
e
x
{\displaystyle e^{x}\,\!}
حالة
a
x
{\displaystyle a^{x}}
عند
a
=e
R
{\displaystyle \mathbb {R} \,\!}
a
x
{\displaystyle a^{x}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
a
x
ln
a
{\displaystyle a^{x}\ln a\,\!}
a
>
0
{\displaystyle a>0\,\!}
R
{\displaystyle \mathbb {R} \,\!}
sin
x
{\displaystyle \sin x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
cos
x
{\displaystyle \cos x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
cos
x
{\displaystyle \cos x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
−
sin
x
{\displaystyle -\sin x\,\!}
R
∖
(
π
2
+
π
Z
)
{\displaystyle \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\,\!}
tan
x
{\displaystyle \tan x\,\!}
R
∖
(
π
2
+
π
Z
)
{\displaystyle \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\,\!}
1
cos
2
x
=
1
+
tan
2
x
{\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x\,\!}
R
∖
(
π
Z
)
{\displaystyle \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)\,\!}
cot
x
{\displaystyle \cot x\,\!}
R
∖
(
π
Z
)
{\displaystyle \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)\,\!}
−
1
sin
2
x
=
−
1
−
cot
2
x
{\displaystyle -{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x\,\!}
[
−
1
,
1
]
{\displaystyle [-1,1]\,\!}
arcsin
x
{\displaystyle \arcsin x\,\!}
]
−
1
,
1
[
{\displaystyle ]-1,1[\,\!}
1
1
−
x
2
{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}\,\!}
[
−
1
,
1
]
{\displaystyle [-1,1]\,\!}
arccos
x
{\displaystyle \arccos x\,\!}
]
−
1
,
1
[
{\displaystyle ]-1,1[\,\!}
−
1
1
−
x
2
{\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
arctan
x
{\displaystyle \arctan x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
1
1
+
x
2
{\displaystyle {\frac {1}{1+x^{2}}}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
sh
x
{\displaystyle \operatorname {sh} x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
ch
x
{\displaystyle \operatorname {ch} x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
ch
x
{\displaystyle \operatorname {ch} x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
sh
x
{\displaystyle \operatorname {sh} x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
th
x
{\displaystyle \operatorname {th} x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
1
ch
2
x
{\displaystyle {\frac {1}{\operatorname {ch} ^{2}x}}\,\!}
R
{\displaystyle \mathbb {R} \,\!}
argsh
x
{\displaystyle \ \operatorname {argsh} \,x\,\!}
R
{\displaystyle \mathbb {R} \,\!}
1
1
+
x
2
{\displaystyle {\frac {1}{\sqrt {1+x^{2}}}}\,\!}
[
1
,
+
∞
[
{\displaystyle [1,+\infty [\,\!}
argch
x
{\displaystyle \ \operatorname {argch} \,x\,\!}
]
1
,
+
∞
[
{\displaystyle ]1,+\infty [\,\!}
1
x
2
−
1
{\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\,\!}
]
−
1
,
1
[
{\displaystyle ]-1,1[\,\!}
argth
x
{\displaystyle \ \operatorname {argth} \,x\,\!}
]
−
1
,
1
[
{\displaystyle ]-1,1[\,\!}
1
1
−
x
2
{\displaystyle {\frac {1}{1-x^{2}}}\,\!}
إذا كانت
g
{\displaystyle g}
إحدى تلك الدوال، فمشتقة
الدالة المركبة
x
↦
g
(
c
x
)
{\displaystyle x\mapsto g(cx)}
(علما أن
c
{\displaystyle c}
عدد حقيقي ثابت) هي
x
↦
c
g
′
(
c
x
)
{\displaystyle x\mapsto cg'(cx)}
.
مقالات ذات صلة
جدول التكاملات
موسوعات ذات صلة :
موسوعة تحليل رياضي
عند 
عند
عند 
عند
![{\displaystyle {\sqrt[{n}]{x}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b680a2f55512c7560a1ab34a646880c1199de2)
![{\displaystyle {\frac {1}{n{\sqrt[{n}]{x^{n-1}}}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8cf1665cab34a3334be5fda96ca4c381b1bcea2)
، حالة عند 
عتد a=e
عند a=e
![{\displaystyle [-1,1]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6adf3a2cce8897cbeab299d7a8bfb9f5505896)
![{\displaystyle ]-1,1[\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cb5252784df43bacaf0fa7ed89a04c275ef200)
![{\displaystyle ]1,+\infty [\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3444b67acf9a5ba2294a9ee211891153669c0f62)
