考研数学笔记 - 求导公式

考研中常用的一些求导公式

Posted by Oscaner on September 9, 2018
\[\begin{align*} ( c )^\prime &= 0 \\ ( x^a )^\prime &= ax^{a - 1} \\ ( \log_a x )^\prime &= \dfrac{1}{x \ln a} \\ ( \ln x )^\prime &= \dfrac{1}{x} \\ ( a^x )^\prime &= x \ln a \\ ( \sin x )^\prime &= \cos x \\ ( \cos x )^\prime &= - \sin x \\ ( \tan x )^\prime &= \dfrac{1}{\cos^2 x} = \sec^2 x \\ ( \cot x )^\prime &= -\dfrac{1}{\sin^2 x} = - \csc^2 x \\ ( \sec x )^\prime &= \dfrac{\sin x}{\cos^2 x} = \sec x \tan x \\ ( \csc x )^\prime &= -\dfrac{\cos x}{\sin^2 x} = -\csc x \cot x \\ ( \arcsin x )^\prime &= \dfrac{1}{\sqrt{1 - x^2}} \\ ( \arccos x )^\prime &= -\dfrac{1}{\sqrt{1 - x^2}} \\ ( \arctan x )^\prime &= \dfrac{1}{1 + x^2} \\ ( \mathrm{arccot} \, x )^\prime &= -\dfrac{1}{1 + x^2} \\ [ \ln ( x + \sqrt{x^2 - 1} )]^\prime &= \dfrac{1}{\sqrt{x^2 - 1}} \\ [ \ln ( x + \sqrt{x^2 + 1} )]^\prime &= \dfrac{1}{\sqrt{x^2 + 1}} \end{align*}\]
\[\begin{align*} ( u \pm v )^{( n )} &= u^{( n )} \pm v^{( n )} \\ ( uv )^{( n )} &= \sum_{k = 0}^{n} \mathrm{C}_{n}^{k} u^{(n - k )} v^{( k )} \end{align*}\]
\[\begin{align*} ( a^x )^{( n )} &= a^x ( \ln a )^n \\ ( e^x )^{( n )} &= e^x \end{align*}\]
\[\begin{align*} [ \sin ( kx ) ]^{( n )} &= k^n \sin ( kx + \dfrac{n}{2} \pi ) \\ [ \cos ( kx ) ]^{( n )} &= k^n \cos ( kx + \dfrac{n}{2} \pi ) \end{align*}\]
\[\begin{align*} ( \ln x )^{( n )} &= ( -1 )^{n - 1} \dfrac{( n - 1 ) !}{x^n} ( x > 0 ) \\ [ \ln ( x + 1 ) ]^{( n )} &= ( -1 )^{n-1} \dfrac{( n - 1 ) !}{( x + 1 )^{n}} ( x > -1 ) \end{align*}\]
\[\begin{align*} ( \dfrac{1}{x + a} )^{( n )} &= ( -1 )^n \dfrac{n !}{( x + a )^{n + 1}} \\ [ ( x + x_0 )^m ]^{( n )} &= m ( m - 1) ( m - 2) \cdots ( m - n + 1 ) ( x + x_0 )^{m - n} \end{align*}\]