$f(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ | $f(x)$ | $F(x)$ |

$f(x)=a$; $a\in \mathbb{R}$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=0$ | $f(x)=a$; $a\in \mathbb{R}$ | $F(x)=ax+C$ |

$f(x)=a{x}^{n}$; $a,n\in \mathbb{R}$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=na{x}^{n-1$ | $f(x)=a{x}^{n}$; $a,n\in \mathbb{R}$, $n\ne -1$ | $F(x)=\frac{{x}^{n+1}}{n+1}+C$ |

$f(x)=\mathrm{ln}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\frac{1}{x$ | $f(x)=\frac{1}{x}={x}^{-1}$ | $F(x)=\mathrm{ln}(x)+C$ |

$f(x)={\mathrm{log}}_{b}(x)$; $b>0,b\ne 1$, $x>0$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\mathrm{ln}(b)$ | $f(x)=\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\mathrm{ln}(b)}$; $b>0,b\ne 1$ | $F(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}={\textstyle \phantom{\rule{-0.167em}{0ex}}}\left({\textstyle \phantom{\rule{-0.167em}{0ex}}}\frac{1}{\mathrm{ln}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(b)}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\right){\textstyle \phantom{\rule{-0.167em}{0ex}}}\mathrm{ln}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}+{\textstyle \phantom{\rule{-0.167em}{0ex}}}C$ |

$f(x)={a}^{x}$; $a>0,a\ne 1$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\mathrm{ln}(a){\textstyle \phantom{\rule{0.167em}{0ex}}}{a}^{x$ | $f(x)={a}^{x}$; $a>0,a\ne 1$ | $F(x)=\frac{{a}^{x}}{\mathrm{ln}(a)}+C$ |

$f(x)={e}^{x}$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)={e}^{x$ | $f(x)={e}^{x}$ | $F(x)={e}^{x}+C$ |

$f(x)=\mathrm{sin}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\mathrm{cos}(x)$ | $f(x)=\mathrm{cos}(x)$ | $F(x)=\mathrm{sin}(x)+C$ |

$f(x)=\mathrm{cos}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=-\mathrm{sin}(x)$ | $f(x)=\mathrm{sin}(x)$ | $F(x)=-\mathrm{cos}(x)+C$ |

$f(x)=\mathrm{tan}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)={\mathrm{sec}}^{2}(x)$ | $f(x)={\mathrm{sec}}^{2}(x)$ | $F(x)=\mathrm{tan}(x)+C$ |

$f(x)=\mathrm{cot}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=-{\mathrm{csc}}^{2}(x)$ | $f(x)={\mathrm{csc}}^{2}(x)$ | $F(x)=-\mathrm{cot}(x)+C$ |

$f(x)=\mathrm{sec}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}={\textstyle \phantom{\rule{-0.167em}{0ex}}}\mathrm{sec}(x)\mathrm{tan}(x)$ | $f(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}={\textstyle \phantom{\rule{-0.167em}{0ex}}}\mathrm{sec}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)\mathrm{tan}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $F(x)=\mathrm{sec}(x)+C$ |

$f(x)=\mathrm{csc}(x)$ | ${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}={\textstyle \phantom{\rule{-0.167em}{0ex}}}-\mathrm{csc}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)\mathrm{cot}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f(x){\textstyle \phantom{\rule{-0.167em}{0ex}}}={\textstyle \phantom{\rule{-0.167em}{0ex}}}\mathrm{csc}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)\mathrm{cot}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $F(x)=-\mathrm{csc}(x)+C$ |

$f(x)={\mathrm{sin}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\frac{1}{\sqrt{1-{x}^{2}}$ | $f(x)=\frac{1}{\sqrt{1-{x}^{2}}}$ | $F(x)={\mathrm{sin}}^{-1}(x)+C$ |

$f(x)={\mathrm{cos}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=-\frac{1}{\sqrt{1-{x}^{2}}$ | $f(x)=\frac{1}{\sqrt{1-{x}^{2}}}$ | $F(x)=-{\mathrm{cos}}^{-1}(x)+C$ |

$f(x)={\mathrm{tan}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\frac{1}{1+{x}^{2}$ | $f(x)=\frac{1}{1+{x}^{2}}$ | $F(x)={\mathrm{tan}}^{-1}(x)+C$ |

$f(x)={\mathrm{cot}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=-\frac{1}{1+{x}^{2}$ | $f(x)=\frac{1}{1+{x}^{2}}$ | $F(x)=-{\mathrm{cot}}^{-1}(x)+C$ |

$f(x)={\mathrm{sec}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\sqrt{{x}^{2}-1}$ | $f(x)=\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\sqrt{{x}^{2}-1}}$ | $F(x)={\mathrm{sec}}^{-1}(x)+C$ |

$f(x)={\mathrm{csc}}^{-1}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)$ | $f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=-\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\sqrt{{x}^{2}-1}$ | $f(x)=\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\sqrt{{x}^{2}-1}}$ | $F(x)=-{\mathrm{csc}}^{-1}(x)+C$ |