常用导数公式

\[ \begin{array}{lllll} \text{三角} & (\tan x)^{\prime} = \sec^2 x & (\cot x)^{\prime} = - \csc^2 x & & \\ & (\sec x)^{\prime} = \sec x \tan x& (\csc x)^{\prime} = - \csc x \cot x & & \\ &(\sin x)^{(n)} = \sin \left( x + \frac{n\pi}{2} \right)&(\cos x)^{(n)} = \cos \left( x + \frac{n\pi}{2} \right)&&\\ \text{反三角} & (\arcsin x)^{\prime} = \frac{1}{\sqrt{1 - x^2}}& (\arccos x)^{\prime} = - \frac{1}{\sqrt{1 - x^2}}& & \\ & (\arctan x)^{\prime} = \frac{1}{1 + x^2}& (\text{arccot} x)^{\prime} = - \frac{1}{1+x^2} & & \\ \text{对数三角}& (\ln |\cos x|)^{\prime} = -\tan x& (\ln |\sin x|)^{\prime} = \cot x& & \\ & (\ln |\sec x + \tan x|)^{\prime} = \sec x& (\ln |\csc x - \cot x|)^{\prime} = \csc x & & \\ \text{对数根式} & (\ln (x + \sqrt{x^2 + a^2}))^{\prime} = \frac{1}{\sqrt{x^2 + a^2}}& (\ln ( x- \sqrt{x^2 + a^2}))^{\prime} = - \frac{1}{\sqrt{x^2 + a^2}}& & \\ & (\ln (x + \sqrt{x^2 - a^2}))^{\prime} = \frac{1}{\sqrt{x^2 - a^2}}& (\ln (x - \sqrt{x^2 - a^2}))^{\prime} = - \frac{1}{\sqrt{x^2 - a^2}} & & \end{array} \]