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1、求导与微分法则

$(c)^{'} =0  \quad dc = 0$

$(cv)^{'}=cv^{'} \quad  d(cv)=cdv$

$(u\pm v)^{'}=u^{'}\pm v^{'} \quad d(u\pm v)=du\pm dv$

$(uv)^{'}=u^{'}v+uv^{'} \quad d(uv)=vdu+udv$

$(\dfrac{u}{v})^{'}=\dfrac{vu^{'}-uv^{'}}{v^2} \quad  d(\dfrac{u}{v})=\dfrac{vdu-udv}{v^2}$

2、导数及微分公式

$(v^n)^{'}=nv^{n-1}v^{'} \quad  dv^n=nv^{n1}dv$

$(\sqrt{v})^{'}=\dfrac{v^{'}}{2\sqrt{v}} \quad  d\sqrt{v}=\frac{dv}{2\sqrt{v}}$

$(\ln v)^{'}=\dfrac{v^{'}}{v} \quad d \ln v = \dfrac{dv}{v}$

$(\log_av)^{'}=\dfrac{v^{'}}{v\ln a} \quad d\log_av=\dfrac{dv}{v\ln a}$

$(e^v)^{'}=e^vv^{'} \quad de^v=e^vdv$

$(a^v)^{'}=a^v\ln a\cdot v^{'} \quad da^v=a^v\ln adv$

$(\sin v)^{'}=\cos v \cdot v^{'} \quad d\sin v=\cos v \cdot dv$

$(\cos v)^{'}=-\sin v \cdot v^{'} \quad d\cos v= -\sin vdv$

$(\tan v)^{'}=\sec^2v\cdot v^{'} \quad  d\tan v=\sec^2 vdv$

$(\cot v)^{'}=-\csc^2v\cdot v^{'} \quad d\cot v=-\csc^2 vdv$

$(\arcsin v)^{'}=\dfrac{v^{'}}{\sqrt{1-v^2}} \quad d\arcsin v = \dfrac{dv}{\sqrt{1-v^2}}$

$(\arccos v)^{'}=-\dfrac{v^{'}}{\sqrt{1-v^2}} \quad d\arccos v=-\dfrac{dv}{\sqrt{1-v^2}}$

$(\arctan v)^{'}=\dfrac{v^{'}}{1+v^2} \quad d\arctan v=\dfrac{dv}{1+v^2}$

3、不定积分表

$\int du= u+C$

$\int u^mdu=\dfrac{u^{m+1}}{m+1}+C$

$\int \dfrac{du}{u} = \ln u+C$

$\int \dfrac{du}{u^2-a^2}=\dfrac{1}{a}\arctan\dfrac{u}{a}+C$

$\int \dfrac{du}{u^2-a^2}=\dfrac{1}{2a}\ln\dfrac{u-a}{u+a}+C$

$\int \dfrac{du}{(u+a)(u-a)}=\dfrac{1}{b-a}\ln\dfrac{u+a}{u+b}+C$

$\int\dfrac{du}{\sqrt{a^2-u^2}}=\arcsin{\dfrac{u}{a}}+C$

$\int e^udu=e^u+C$

$\int a^udu=\dfrac{a^u}{\ln a}+C$

$\int \sin udu = -\cos u+C$

$\int \cos udu = \sin u +C$

$\int \tan udu = -\ln \cos u$

$\int \cot udu = \ln \sin u +C$

$\int \sec^2 udu = \int \dfrac{du}{\cos^2 u}=\tan u+C$

$\int\csc^2 udu=\int \dfrac{du}{\sin^2 u}=\cot u+C$

$\int \sec udu = \int \dfrac{du}{\cos u}=\ln(\sec u+\tan u)+C = \ln \tan(\dfrac{u}{2}+\dfrac{\pi}{4})+C$

$\int \csc udu = \int \dfrac{du}{\sin u} = \ln(\csc u -\cot u)+C = \ln \tan \dfrac{u}{2}+C$

$\int \sec u\tan udu = \sec u+C$

$\int \csc u \cot udu = -\csc u+C$