مشتقات عادية

مشتقات الدوال الاعتيادية :
| مجال الدالة  D_{f}\,\! | الدالة  f(x)\,\! | مجال المشتقة  D_{f'}\,\! | المشتقة  f'(x)\,\! | تعليق |
| --- | --- | --- | --- | --- |
|  \mathbb {R} \,\! |  k\,\! |  \mathbb {R} \,\! |  0\,\! |  k\in \mathbb {R}  |
|  \mathbb {R} \,\! |  x\,\! |  \mathbb {R} \,\! |  1\,\! | حال  x^{n} عند  n=1 |
|  \mathbb {R} \,\! |  x^{2}\,\! |  \mathbb {R} \,\! |  2x\,\! | حالة  n=2 عند  x^{n} |
|  \mathbb {R} _{+}\,\! |  {\sqrt {x}}\,\! |  \mathbb {R} _{+}^{*}\,\! |  {\frac {1}{2{\sqrt {x}}}}\,\! | حالة  x^{\alpha } عند  \alpha =1/2 |
|  \mathbb {R} ^{*}\,\! |  {\frac {1}{x}}\,\! |  \mathbb {R} ^{*}\,\! |  -{\frac {1}{x^{2}}}\,\! | حالة  1/x^{n} عند  n=1 |
|  \mathbb {R} \,\! |  x^{n}\,\! |  \mathbb {R} \,\! |  nx^{n-1}\,\! |  n\in \mathbb {N} \,\! |
|  \mathbb {R} ^{*}\,\! |  {\frac {1}{x^{n}}}\,\! |  \mathbb {R} ^{*}\,\! |  -{\frac {n}{x^{n+1}}}\,\! |  n\in \mathbb {N} \,\! |
|  \mathbb {R} _{+}\,\! |  {\sqrt[{n}]{x}}\,\! |  \mathbb {R} _{+}^{*}\,\! |  {\frac {1}{n{\sqrt[{n}]{x^{n-1}}}}}\,\! |  n\in \mathbb {N} ~، حالة   x^{\alpha } عند  \alpha =1/n |
|  \mathbb {R} _{+}\,\! |  x^{\alpha }\,\! |  \mathbb {R} _{+}\,\! |  \alpha x^{\alpha -1}\,\! |  \alpha \geq 1\,\! |
|  \mathbb {R} _{+}\,\! |  x^{\alpha }\,\! |  \mathbb {R} _{+}^{*}\,\! |  \alpha x^{\alpha -1}\,\! |  0<\alpha <1\,\! |
|  \mathbb {R} _{+}^{*}\,\! |  x^{\alpha }\,\! |  \mathbb {R} _{+}^{*}\,\! |  \alpha x^{\alpha -1}\,\! |  \alpha <0\,\! |
|  \mathbb {R} ^{*}\,\! |  \ln |x|\,\! |  \mathbb {R} ^{*}\,\! |  {\frac {1}{x}}\,\! | حالة  \log _{a}x عتد a=e |
|  \mathbb {R} ^{*}\,\! |  \log _{a}|x|\,\! |  \mathbb {R} ^{*}\,\! |  {\frac {1}{x\ln a}}\,\! |  a>0\,\! |
|  \mathbb {R} \,\! |  e^{x}\,\! |  \mathbb {R} \,\! |  e^{x}\,\! | حالة  a^{x} عند a=e |
|  \mathbb {R} \,\! |  a^{x}\,\! |  \mathbb {R} \,\! |  a^{x}\ln a\,\! |  a>0\,\! |
|  \mathbb {R} \,\! |  \sin x\,\! |  \mathbb {R} \,\! |  \cos x\,\! |  |
|  \mathbb {R} \,\! |  \cos x\,\! |  \mathbb {R} \,\! |  -\sin x\,\! |  |
|  \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\,\! |  \tan x\,\! |  \mathbb {R} \backslash \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\,\! |  {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x\,\! |  |
|  \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)\,\! |  \cot x\,\! |  \mathbb {R} \backslash \left(\pi \mathbb {Z} \right)\,\! |  -{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x\,\! |  |
|  [-1,1]\,\! |  \arcsin x\,\! |  ]-1,1[\,\! |  {\frac {1}{\sqrt {1-x^{2}}}}\,\! |  |
|  [-1,1]\,\! |  \arccos x\,\! |  ]-1,1[\,\! |  -{\frac {1}{\sqrt {1-x^{2}}}}\,\! |  |
|  \mathbb {R} \,\! |  \arctan x\,\! |  \mathbb {R} \,\! |  {\frac {1}{1+x^{2}}}\,\! |  |
|  \mathbb {R} \,\! |  \operatorname {sh} x\,\! |  \mathbb {R} \,\! |  \operatorname {ch} x\,\! |  |
|  \mathbb {R} \,\! |  \operatorname {ch} x\,\! |  \mathbb {R} \,\! |  \operatorname {sh} x\,\! |  |
|  \mathbb {R} \,\! |  \operatorname {th} x\,\! |  \mathbb {R} \,\! |  {\frac {1}{\operatorname {ch} ^{2}x}}\,\! |  |
|  \mathbb {R} \,\! |  \ \operatorname {argsh} \,x\,\! |  \mathbb {R} \,\! |  {\frac {1}{\sqrt {1+x^{2}}}}\,\! |  |
|  [1,+\infty [\,\! |  \ \operatorname {argch} \,x\,\! |  ]1,+\infty [\,\! |  {\frac {1}{\sqrt {x^{2}-1}}}\,\! |  |
|  ]-1,1[\,\! |  \ \operatorname {argth} \,x\,\! |  ]-1,1[\,\! |  {\frac {1}{1-x^{2}}}\,\! |  |