يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. (مارس 2016) هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. (نوفمبر 2014) مشتقات الدوال الاعتيادية : مزيد من المعلومات مجال الدالة ... 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}}}\,\!} إغلاق 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)} . جدول التكاملات بوابة تحليل رياضي هذه بذرة مقالة عن التحليل الرياضي بحاجة للتوسيع. فضلًا شارك في تحريرها. 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Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.