导数

常用导数

1. $(\tan x{)}^{\prime }={\sec }^{2}x$

证明

$\begin{array}{rl}{\tan x}^{\prime }& ={\left(\frac{\sin x}{\cos x}\right)}^{\prime }\\ \\ & =\frac{(\sin x{)}^{\prime }\cos x-\sin x(\cos x{)}^{\prime }}{{\cos }^{2}x}\\ \\ & =\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}\\ \\ & =\frac{1}{{\cos }^{2}x}\\ \\ & ={\sec }^{2}x& \\ \end{array}$

2. $(\sec x{)}^{\prime }=\tan x\sec x$

证明

$\begin{array}{rl}{\sec x}^{\prime }& ={\left(\frac{1}{\cos x}\right)}^{\prime }\\ \\ & =\frac{1^{\prime }\cdot \cos x-1\cdot (\cos x{)}^{\prime }}{{\cos }^{2}x}\\ \\ & =\frac{-(\cos x{)}^{\prime }}{{\cos }^{2}x}\\ \\ & =\frac{\sin x}{{\cos }^{2}x}\\ \\ & =\frac{\sin x}{\cos x}\cdot \frac{1}{\cos x}\\ \\ & =\tan x\sec x& \\ \end{array}$