基本求导法则与导数公式

基本求导法则和基本初等函数的导数公式在初等函数的求导运算中起着重要的作用，需要熟练掌握，现在归纳如下，我们可以利用本章的定理和例题的证明过程来记忆：

1. 常数和基本初等函数的导数公式

$(1) (C)'=0$	$(2)\left( {{x^\mu }} \right)' = \mu {x^{\mu - 1}}$
$(3)\left( {\sin x} \right)' = \cos x$	$\left( 4 \right)\left( {\cos x} \right)' = - \sin x$
$\left( 5 \right)(\tan x)' = {\text{se}}{{\text{c}}^2}x$	$\left( 6 \right)\left( {\cot x} \right)' = - {\text{cs}}{{\text{c}}^2}x$
$\left( 7 \right)\left( {\sec x} \right)' = \sec x\tan x$	$\left( 8 \right)\left( {\csc x} \right)' = - \csc x\cot x$
$\left( 9 \right)({a^x})' = {a^x}\ln a$	$(10)\left( {{{{\bf{e}}}^x}} \right)' = {{{\bf{e}}}^x}$
$(11)\left( {{{\log }_a}x} \right)' = \frac{1}{{x\ln a}}$	$(12)\left( {\ln x} \right)' = \frac{1}{x}$
$$(13)\left( {\arcsin x} \right)' = \frac{1}{{\sqrt {1 - {x^2}} }}$$
$$(14)\left( {\arccos x} \right)' = - \frac{1}{{\sqrt {1 - {x^2}} }}$$
$$(15)\left( {\arctan x} \right)' = \frac{1}{{1 + {x^2}}}$$
$$(16)\left( {{\text{arccot}} x} \right)' = - \frac{1}{{1 + {x^2}}}$$

2. 函数的和、差、积、商的求导法则

设$u=u(x),v=v(x)$都可导，则

$$(1) (u±v)'=u'±v'$$ $$(2) (Cu)'=Cu' (C是常数)$$ $$(3) (uv)'=u' v+uv' $$ $$(4)\left( {\frac{u}{v}} \right)' = \frac{{u'v - uv'}}{{{v^2}}}$$

3. 反函数的求导法则

设$x=f(y)$在区间$I_y$内单调、可导且$f'(y)≠0$，则它的反函数$y=f^{-1} (x)$在区间$I_x=f(I_y)$内也可导，且

$$\left[ {{f^{ - 1}}\left( x \right)} \right]' = \frac{1}{{f'\left( y \right)}}$$

或

$$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{{\frac{{{\text{d}}x}}{{{\text{d}}y}}}}$$

4. 复合函数的求导法则

设$y=f(u)$，而$u=g(x)$且$f(u)$及$g(x)$都可导，则复合函数$y=f[g(x)]$的导数为

$$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \cdot \frac{{{\text{d}}u}}{{{\text{d}}x}}$$

或

$$y'\left( x \right) = f'(u) \cdot g'(x)$$