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