导数
可导性判定
- $f(x)$在$x_0$可导的充分条件为：$f(x)$在$x_0$左右可导并有 $f'_-(x)=f'_+(x)$
- 可导一定连续，连续不一定可导
链式法则求导
设$y=f(x)$,则$f'(t)=frac{{ m d}y}{{ m d}x}=frac{{ m d}y}{{ m d}t}frac{{ m d}t}{{ m d}x}$
常用于复杂的复合函数求导
$eg.1求(a^x)'$
$ax=e{xlna}
ightarrow 令u=xlna Rightarrow$
$$
frac{{
m d}a^x}{{
m d}x}=frac{{
m d}e^u}{{
m d}u}frac{{
m d}u}{{
m d}x}=eulna=axlna
$$
$eg.2设y=sinsqrt{x^2+x+1},求y'$
$设u=x^2+x+1,v=sqrt{u}，由链式法则Rightarrow $
$$frac{{
m d}y}{{
m d}x}=frac{{
m d}y}{{
m d}v}frac{{
m d}v}{{
m d}u}frac{{
m d}u}{{
m d}x}=cosvfrac{1}{2sqrt{u}}(2x+1)=frac{2x+1}{2sqrt{x2+x+1}}cossqrt{x2+x+1}$$
对数求导
$y=f(x) ightarrow lny=lnf(x) ightarrow (lny)'=(lnf(x))'Rightarrow$
$y'frac{1}{y}=(lnf(x))'Rightarrow y'=y(lnf(x))'$
$eg.3设y=sqrt[3]{frac{x2(x-1)(x-2)}{(x+1)(x+3)2}},求y'$
$egin{aligned}
y' & =y(lnfrac{x2(x-1)(x-2)}{(x+1)(x+3)2})' \
& = y(frac{2}{3}lnx+frac{1}{3}ln(x-1)+frac{1}{3}ln(x+2)-frac{1}{3}ln(x+1)-frac{2}{3}ln(x+3))' \
& = sqrt[3]{frac{x2(x-1)(x-2)}{(x+1)(x+3)2}}(frac{2}{3(x^2+1)}+frac{2}{x(x+3)}+frac{1}{3(x+2)}) \
end{aligned}
$
对数求导用于函数中同时包含幂函数、乘法、除法的复杂复合函数
反函数求导
$由链式法则可得：frac{{
m d}y}{{
m d}x}=frac{1}{frac{{
m d}x}{{
m d}y}}$
$eg.4设x=arcsiny,求x'$
$x=arcsiny
ightarrow y=sinxRightarrow $
$$
frac{{
m d}x}{{
m d}y} = frac{1}{frac{{
m d}y}{{
m d}x}} = frac{1}{frac{{
m d}sinx}{{
m d}x}} = frac{1}{cosx} = frac{1}{sqrt{1-sin2x}}=frac{1}{sqrt{1-y2}}
$$
幂指函数求导
先将底化为常数，再对指数进行求导
$eg.5设y=x^x求y'$
$y'=(e{xlnx})'=(xlnx)'e{xlnx}=(lnx+1)x^x$
高阶导数
$(sinx)^{(n)}=sin(frac{npi}{2}+x)$
$(cosx)^{(n)}=cos(frac{npi}{2}+x)$
$ln(1+x){(n)}=frac{(-1){n-1}(n-1)!}{(1+x)^n}$
$sin(ax+b){(n)}=ansin(ax+b+frac{npi}{2})$
$(e{ax+b}){(n)}=ane{ax+b}$
$[u(x)v(x)]{(n)}=sum_{k=0}{n}C_{n}{k}u{(k)}v^{(n-k)}$