# 微积分基本函数

## 基本函数表

\displaystyle \frac{x^{n+1}}{x+1}\Longrightarrow x^n\Longrightarrow nx^{n-1}\\{}\\ -\cos x\Longrightarrow\sin x\Longrightarrow\cos x\\{}\\ \sin x\Longrightarrow\cos x\Longrightarrow-\sin x\\{}\\ \frac{e^{cx}}{c}\Longrightarrow e^{cx}\Longrightarrow ce^{cx}\\{}\\ x\ln x-x\Longrightarrow\ln x\Longrightarrow\frac{1}{x}

## 1/f(x) 的求导

\displaystyle \left(\frac{1}{f(x)}\right)^\prime=[f(x)^{-1}]^\prime=-f^{-2}\cdot f^\prime

\displaystyle \left(\frac{1}{f}\right)^\prime=g^\prime f^\prime=-f^{-2}f^\prime

\displaystyle \left(\frac{1}{v}\right)^\prime=(v^{-1})^\prime=-v^{-2}\cdot v^\prime

\displaystyle \left(\frac{1}{x}\right)^\prime=(x^{-1})^\prime=-x^{-2}\cdot x^\prime\\{}\\ =-x^{-2}\cdot1=-x^{-2}

\displaystyle \left(\frac{u}{v}\right)^\prime=(uv^{-1})^\prime\\{}\\ =u^\prime v^{-1}+u(-v^{-2}v^\prime)\\{}\\ =\frac{u^\prime}{v}-\frac{uv^\prime}{v^2}\\{}\\ =\frac{u^\prime v-uv^\prime}{v^2}

## ln三角函数的求导

\displaystyle (\sec x)^\prime=\frac{\mathrm{d}}{\mathrm{d}x}(\cos x)^{-1}\\{}\\ =-(\cos x)^{-2}(-\sin x)\\{}\\ =\frac{\sin x}{\cos^2x}=\frac{1}{\cos x}\cdot\frac{\sin x}{\cos x}=\sec x\tan x\\{}\\ \frac{\mathrm{d}}{\mathrm{d}x}\ln(\sec x)=\frac{(\sec x)^\prime}{\sec x}\\{}\\ =\frac{\sec x\tan x}{\sec x}\\{}\\ =\tan x\\{}\\ \frac{\mathrm{d}}{\mathrm{d}x}e^{x\tan^{-1}x}=e^{x\tan^{-1}x}\frac{\mathrm{d}}{\mathrm{d}x}(x\tan^{-1}x)\\{}\\ =e^{x\tan^{-1}x}\left(\tan^{-1}x+\frac{x}{1+x^2}\right)