考研数学笔记 – 求导公式
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$$ \left( c \right)^\prime = 0 $$

$$ \left( x^a \right)^\prime = ax^{a - 1}$$

$$ \left( \log_a x \right)^\prime = \dfrac{1}{x \ln a} $$

$$ \left( \ln x \right)^\prime = \dfrac{1}{x} $$

$$ \left( a^x \right)^\prime = x \ln a $$

$$ \left( \sin x \right)^\prime = \cos x $$

$$ \left( \cos x \right)^\prime = - \sin x $$

$$ \left( \tan x \right)^\prime = \dfrac{1}{\cos^2 x} = \sec^2 x $$

$$ \left( \cot x \right)^\prime = -\dfrac{1}{\sin^2 x} = - \csc^2 x $$

$$ \left( \sec x \right)^\prime = \dfrac{\sin x}{\cos^2 x} = \sec x \tan x $$

$$ \left( \csc x \right)^\prime = -\dfrac{\cos x}{\sin^2 x} = -\csc x \cot x $$

$$ \left( \arcsin x \right)^\prime = \dfrac{1}{\sqrt{1 - x^2}} $$

$$ \left( \arccos x \right)^\prime = -\dfrac{1}{\sqrt{1 - x^2}} $$

$$ \left( \arctan x \right)^\prime = \dfrac{1}{1 + x^2} $$

$$ \left( \mathrm{arccot} \, x \right)^\prime = -\dfrac{1}{1 + x^2} $$

$$ \left[ \ln \left( x + \sqrt{x^2 - 1} \right)\right]^\prime = \dfrac{1}{\sqrt{x^2 - 1}} $$

$$ \left[ \ln \left( x + \sqrt{x^2 + 1} \right)\right]^\prime = \dfrac{1}{\sqrt{x^2 + 1}} $$


$$ \left( u \pm v \right)^{\left( n \right)} = u^{\left( n \right)} \pm v^{\left( n \right)} $$

$$ \left( uv \right)^{( n )} = \sum_{k = 0}^{n} \mathrm{C}_{n}^{k} u^{\left(n - k \right)} v^{\left( k \right)} $$


$$ \left( a^x \right)^{\left( n \right)} = a^x \left( \ln a \right)^n $$

$$ \left( e^x \right)^{\left( n \right)} = e^x $$


$$ \left[ \sin \left( kx \right) \right]^{\left( n \right)} = k^n \sin \left( kx + \dfrac{n}{2} \pi \right) $$

$$ \left[ \cos \left( kx \right) \right]^{\left( n \right)} = k^n \cos \left( kx + \dfrac{n}{2} \pi \right) $$


$$ \left( \ln x \right)^{\left( n \right)} = \left( -1 \right)^{n - 1} \dfrac{\left( n - 1 \right) !}{x^n} \left( x > 0 \right) $$

$$ \left[ \ln \left( x + 1 \right) \right]^{\left( n \right)} = \left( -1 \right)^{n-1} \dfrac{\left( n - 1 \right) !}{\left( x + 1 \right)^{n}} \left( x > -1 \right) $$


$$ \left( \dfrac{1}{x + a} \right)^{\left( n \right)} = \left( -1 \right)^n \dfrac{n !}{\left( x + a \right)^{n + 1}} $$

$$ \left[ \left( x + x_0 \right)^m \right]^{\left( n \right)} = m \left( m - 1\right) \left( m - 2\right) \cdots \left( m - n + 1 \right) \left( x + x_0 \right)^{m - n} $$

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