• $(x^{\alpha }{)}^{\prime }=\alpha x^{\alpha -1}$
• $(e^x)=e^x$
  • $(a^x{)}^{\prime }=a^x\ln a$
• $(\ln x{)}^{\prime }={\displaystyle \frac{1}{x}}$
  • 进一步有$(\ln |x|{)}^{\prime }={\displaystyle \frac{1}{x}}$
  • $({\log }_{a}x{)}^{\prime }={\displaystyle \frac{1}{\ln a\cdot x}}$
• $(\sin x{)}^{\prime }=\cos x$
• $(\cos x{)}^{\prime }=-\sin x$
• $(\tan x{)}^{\prime }={\sec }^{2}x$
• $(\cot x{)}^{\prime }=-{\csc }^{2}x$
• $(\arcsin x{)}^{\prime }={\displaystyle \frac{1}{\sqrt{1-x^2}}}$
• $(\arccos x{)}^{\prime }=-{\displaystyle \frac{1}{\sqrt{1-x^2}}}$
• $(\arctan x{)}^{\prime }={\displaystyle \frac{1}{1+x^2}}$
• $(\mathrm{arccot}x{)}^{\prime }=-{\displaystyle \frac{1}{1+x^2}}$

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• ${(\ln (x+\sqrt{x^2+a^2}))}^{\prime }={\displaystyle \frac{1}{\sqrt{x^2+a^2}}}$
• ${(\ln |x-\sqrt{x^2+a^2}|)}^{\prime }={\displaystyle \frac{1}{\sqrt{x^2-a^2}}}$

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• $(x^x{)}^{\prime }=x^x(\ln x+1)$
  • $(x^x{)}^{\prime }=(e^{x\ln x}{)}^{\prime }$