微積分公式表

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微分公式 積分公式
  1. \begin{matrix}\frac{d}{dx}[cu]=cu' \end{matrix}
  2. \begin{matrix}\frac{d}{dx}[u\pm v]=u'\pm v\end{matrix}
  3. \begin{matrix}\frac{d}{dx}[uv]=uv'+vu' \end{matrix}
  4. \begin{matrix}\frac{d}{dx}[ \frac{u}{v}]=\frac{vu'-uv'}{v^2} \end{matrix}
  5. \begin{matrix}\frac{d}{dx}[c]=0 \end{matrix}
  6. \begin{matrix}\frac{d}{dx}[u^n]=nu^{n-1}u' \end{matrix}
  7. \begin{matrix}\frac{d}{dx}[x]=1 \end{matrix}
  8. \begin{matrix}\frac{d}{dx}[|u|]=\frac{u}{|u|}(u'),u\ne 0 \end{matrix}
  9. \begin{matrix}\frac{d}{dx}[\ln u\,]=\frac{u'}{u} \end{matrix}
  10. \begin{matrix}\frac{d}{dx}[e^u]=e^uu' \end{matrix}
  11. \begin{matrix}\frac{d}{dx}[\log_au]=\frac{u'}{(\ln a)u} \end{matrix}
  12. \begin{matrix}\frac{d}{dx}[a^u]=(\ln a)a^uu' \end{matrix}
  13. \begin{matrix}\frac{d}{dx}[\sin u]=(\cos u)u' \end{matrix}
  14. \begin{matrix}\frac{d}{dx}[\cos u]=-(\sin u)u' \end{matrix}
  15. \begin{matrix}\frac{d}{dx}[\tan u]=(\sec ^2u)u' \end{matrix}
  16. \begin{matrix}\frac{d}{dx}[\cot u]=-(\csc ^2u)u' \end{matrix}
  17. \begin{matrix}\frac{d}{dx}[\sec u]=(\sec u\tan u)u' \end{matrix}
  18. \begin{matrix}\frac{d}{dx}[\csc u]=-(\csc u\cot u)u' \end{matrix}
  19. \begin{matrix}\frac{d}{dx}[\arcsin u]=\frac{u'}{\sqrt{1-u^2}} \end{matrix}
  20. \begin{matrix}\frac{d}{dx}[\arccos u]=\frac{-u'}{\sqrt{1-u^2}} \end{matrix}
  21. \begin{matrix}\frac{d}{dx}[\arctan u]=\frac{u'}{\sqrt{1+u^2}} \end{matrix}
  22. \begin{matrix}\frac{d}{dx}[\arccot u]=\frac{-u'}{1+u^2} \end{matrix}
  23. \begin{matrix}\frac{d}{dx}[\arcsec u]=\frac{u'}{|u|\sqrt{u^2-1}} \end{matrix}
  24. \begin{matrix}\frac{d}{dx}[\arccsc u]=\frac{-u'}{|u|\sqrt{u^2-1}} \end{matrix}
  25. \begin{matrix}\frac{d}{dx}[\sinh u]=(\cosh u)u' \end{matrix}
  26. \begin{matrix}\frac{d}{dx}[\cosh u]=(\sinh u)u' \end{matrix}
  27. \begin{matrix}\frac{d}{dx}[\tanh u]=(\mbox{sech}^2u)u' \end{matrix}
  28. \begin{matrix}\frac{d}{dx}[\coth u]=-(\mbox{csch}^2u)u' \end{matrix}
  29. \begin{matrix}\frac{d}{dx}[\mbox{sech}\ u]=-(\mbox{sech}\ u\tanh u)u' \end{matrix}
  30. \begin{matrix}\frac{d}{dx}[\mbox{csch}\ u]=-(\mbox{csch}\ u\coth u)u' \end{matrix}
  31. \begin{matrix}\frac{d}{dx}[\sinh ^{-1}u]=\frac{u'}{\sqrt{u^2+1}} \end{matrix}
  32. \begin{matrix}\frac{d}{dx}[\cosh ^{-1}u]=\frac{u'}{\sqrt{u^2-1}} \end{matrix}
  33. \begin{matrix}\frac{d}{dx}[\tanh ^{-1}u]=\frac{u'}{1-u^2} \end{matrix}
  34. \begin{matrix}\frac{d}{dx}[\coth ^{-1}u]=\frac{u'}{1-u^2} \end{matrix}
  35. \begin{matrix}\frac{d}{dx}[\mbox{sech}^{-1}\ u]=\frac{-u'}{u\sqrt{1-u^2}} \end{matrix}
  36. \begin{matrix}\frac{d}{dx}[\mbox{csch}^{-1}\ u]=\frac{-u'}{|u|\sqrt{1+u^2}} \end{matrix}
  1. \begin{matrix}\int k\!f\!(\!u\!)\ du=k\!\int\!f\!(\!u\!)\ du \end{matrix}
  2. \begin{matrix}\int [f\!(\!u\!)\pm g\!(\!u\!)]du=\int f\!(\!u\!)du\pm \int \!g\!(\!u\!)du \end{matrix}
  3. \begin{matrix}\int du=u+C \end{matrix}
  4. \begin{matrix}\int a^u\,du=(\frac{1}{\ln a})a^u+C \end{matrix}
  5. \begin{matrix}\int e^u\,du=e^u+C \end{matrix}
  6. \begin{matrix}\int \sin u\,du=-\cos u+C \end{matrix}
  7. \begin{matrix}\int \cos u\,du=\sin u+C \end{matrix}
  8. \begin{matrix}\int \tan u\,du=-\ln|\cos u|+C \end{matrix}
  9. \begin{matrix}\int \cot u\,du=\ln|\sin u|+C \end{matrix}
  10. \begin{matrix}\int \sec u\,du=\ln|\sec u+\tan u|+C \end{matrix}
  11. \begin{matrix}\int \csc u\,du=-\ln|\csc u+\cot u|+C \end{matrix}
  12. \begin{matrix}\int \sec^2u\,du=\tan u+C \end{matrix}
  13. \begin{matrix}\int \csc^2u\,du=-\cot u+C \end{matrix}
  14. \begin{matrix}\int \sec u\tan u\,du=\sec u+C \end{matrix}
  15. \begin{matrix}\int \csc u\cot u\,du=-\csc u+C \end{matrix}
  16. \begin{matrix}\int \frac{du}{\sqrt{a^2-u^2}}=\mbox{arcsin}\frac{u}{a}+C \end{matrix}
  17. \begin{matrix}\int \frac{du}{a^2+u^2}=\frac{1}{a}\mbox{arctan}\frac{u}{a}+C \end{matrix}
  18. \begin{matrix}\int \frac{du}{u\sqrt{u^2-a^2}}=\frac{1}{a}\mbox{arcsec}\frac{|u|}{a}+C \end{matrix}
取自"http://eid.niu.edu.tw/niuwiki/index.php/%E5%BE%AE%E7%A9%8D%E5%88%86%E5%85%AC%E5%BC%8F%E8%A1%A8"
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