### Product Rule $ \frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x) $ ### Quotient Rule $ \frac{d}{dx}\left( \frac{f(x)}{g(x)} \right)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}} $ ### Chain Rule $ \frac{d}{dx}f(g(x))=f'(g(x))g'(x) $ ### Derivatives | Function | Derivative | | :-----------------------: | :--------------------------------------: | | $c$ | $0$ | | $x$ | $1$ | | $x^n$ | $nx^{n-1}$ | | $g(u(x))$ | $\dfrac{dg}{du}\,\dfrac{du}{dx}$ | | $u+v$ | $u'+v'$ | | $uv$ | $uv'+vu'$ | | $\dfrac{u}{v}$ | $\dfrac{vu'-uv'}{v^2}$ | | $\sin x$ | $\cos x$ | | $\cos x$ | $-\sin x$ | | $\tan x$ | $\sec^2 x$ | | $\cot x$ | $-\csc^2 x$ | | $\sec x$ | $\sec x\tan x$ | | $\csc x$ | $-\csc x\cot x$ | | $e^x$ | $e^x$ | | $\ln x$ | $\dfrac{1}{x}$ | | $a^x$ | $a^x\ln a$ | | $\log_{10} x$ | $\dfrac{1}{x\ln 10}$ | | $\sqrt{x}$ | $\dfrac{1}{2\sqrt{x}}$ | | $\arcsin x$ | $\dfrac{1}{\sqrt{1-x^2}}$ | | $\arccos x$ | $-\dfrac{1}{\sqrt{1-x^2}}$ | | $\arctan x$ | $\dfrac{1}{1+x^2}$ | | $\operatorname{arccot} x$ | $-\dfrac{1}{1+x^2}$ | | $\operatorname{arcsec} x$ | $\dfrac{1}{\lvert x\rvert\sqrt{x^2-1}}$ | | $\operatorname{arccsc} x$ | $-\dfrac{1}{\lvert x\rvert\sqrt{x^2-1}}$ | | $\ln\sin x$ | $\cot x$ | | $\ln\cos x$ | $-\tan x$ | | $\ln\tan x$ | $\dfrac{2}{\sin 2x}$ | | $\ln\cot x$ | $-\dfrac{2}{\sin 2x}$ | ### Integrals | Integral | Antiderivative | Condition | | :---: | :---: | :---: | | $\displaystyle\int c\,dx$ | $cx$ | | | $\displaystyle\int x^n\,dx$ | $\dfrac{x^{n+1}}{n+1}$ | $n\neq-1$ | | $\displaystyle\int \dfrac{dx}{x}$ | $\ln\lvert x\rvert$ | | | $\displaystyle\int \dfrac{dx}{ax+b}$ | $\dfrac{1}{a}\ln\lvert ax+b\rvert$ | $a\neq 0$ | | $\displaystyle\int \dfrac{dx}{\sqrt{x}}$ | $2\sqrt{x}$ | | | $\displaystyle\int (u\pm v)\,dx$ | $\displaystyle\int u\,dx\pm\int v\,dx$ | | | $\displaystyle\int u\,dv$ | $uv-\displaystyle\int v\,du$ | by parts | | $\displaystyle\int e^x\,dx$ | $e^x$ | | | $\displaystyle\int a^x\,dx$ | $\dfrac{a^x}{\ln a}$ | $a>0,\ a\neq1$ | | $\displaystyle\int \ln x\,dx$ | $x\ln x-x$ | | | $\displaystyle\int \sin x\,dx$ | $-\cos x$ | | | $\displaystyle\int \cos x\,dx$ | $\sin x$ | | | $\displaystyle\int \tan x\,dx$ | $-\ln\lvert\cos x\rvert$ | | | $\displaystyle\int \cot x\,dx$ | $\ln\lvert\sin x\rvert$ | | | $\displaystyle\int \dfrac{dx}{\sin x}$ | $\ln\left\lvert\tan\dfrac{x}{2}\right\rvert$ | | | $\displaystyle\int \dfrac{dx}{\cos x}$ | $\ln\left\lvert\tan\!\left(\dfrac{\pi}{4}+\dfrac{x}{2}\right)\right\rvert$ | | | $\displaystyle\int \sin^2 x\,dx$ | $\dfrac{x}{2}-\dfrac{\sin 2x}{4}$ | | | $\displaystyle\int \cos^2 x\,dx$ | $\dfrac{x}{2}+\dfrac{\sin 2x}{4}$ | | | $\displaystyle\int \dfrac{dx}{1+\cos x}$ | $\tan\dfrac{x}{2}$ | | | $\displaystyle\int \dfrac{dx}{\sqrt{b^2-x^2}}$ | $\arcsin\dfrac{x}{b}$ | $b>0$ | | $\displaystyle\int \dfrac{dx}{\sqrt{x^2-b^2}}$ | $\ln\left\lvert x+\sqrt{x^2-b^2}\right\rvert$ | | | $\displaystyle\int \dfrac{dx}{b^2+x^2}$ | $\dfrac{1}{b}\arctan\dfrac{x}{b}$ | | | $\displaystyle\int \dfrac{dx}{b^2-x^2}$ | $\dfrac{1}{2b}\ln\left\lvert\dfrac{x+b}{x-b}\right\rvert$ | | | $\displaystyle\int \dfrac{dx}{x^2-b^2}$ | $\dfrac{1}{2b}\ln\left\lvert\dfrac{x-b}{x+b}\right\rvert$ | | | $\displaystyle\int \dfrac{dx}{ax^2+bx+c}$ | $\dfrac{2}{\sqrt{4ac-b^2}}\arctan\dfrac{2ax+b}{\sqrt{4ac-b^2}}$ | $4ac>b^2$ | | $\displaystyle\int e^{ax}\sin bx\,dx$ | $\dfrac{e^{ax}(a\sin bx-b\cos bx)}{a^2+b^2}$ | | | $\displaystyle\int e^{ax}\cos bx\,dx$ | $\dfrac{e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}$ | |