微积分学/导数的概念

定义

一般定义

${\displaystyle f'(x_{0})=\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}=\lim _{\Delta x\to 0}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$

${\displaystyle f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

几何意义

${\displaystyle \tan \varphi ={\frac {\Delta y}{\Delta x}}={\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$

${\displaystyle P_{0}}$处的切线${\displaystyle P_{0}T}$，即${\displaystyle PP_{0}}$的极限位置存在时，此时${\displaystyle \Delta x\to 0}$${\displaystyle \varphi \to \alpha }$，则${\displaystyle P_{0}T}$的斜率${\displaystyle \tan \alpha }$为：

${\displaystyle \tan \alpha =\lim _{\Delta x\to 0}\tan \varphi =\lim _{\Delta x\to 0}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$

函数可导的条件

${\displaystyle \lim _{\Delta x\to 0}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}=\lim _{\Delta x\to 0^{-}}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}=\lim _{\Delta x\to 0^{+}}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$

 左导数：${\displaystyle f'_{-}(x_{0})=\lim _{\Delta x\to 0^{-}}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$
 右导数：${\displaystyle f'_{+}(x_{0})=\lim _{\Delta x\to 0^{+}}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}}$

1.上面这个符号函数在${\displaystyle x=0}$处可导吗？

2.上面这个绝对值函数在${\displaystyle x=0}$处可导吗？

导数的求导法则

四则运算的求导法则

 求导法则 1 ${\displaystyle [u(x)\pm v(x)]'=u'(x)\pm v'(x)}$ 2 ${\displaystyle [u(x)v(x)]'=u'(x)v(x)+u(x)v'(x)}$ 3 ${\displaystyle \left[{\frac {u(x)}{v(x)}}\right]'={\frac {u'(x)v(x)-u(x)v'(x)}{v^{2}(x)}}}$

 4 ${\displaystyle [Cv(x)]'=Cv'(x)}$ 5 ${\displaystyle \left[{\frac {C}{v(x)}}\right]'={\frac {-Cv'(x)}{v^{2}(x)}}}$

• 证明${\displaystyle [u(x)v(x)]'=u'(x)v(x)+u(x)v'(x)}$

复合函数求导

 求导法则 1 ${\displaystyle \{u[v(x)]\}'=u'[v(x)]v'(x)}$

反函数的求导

${\displaystyle [f^{-1}(y)]'={\frac {1}{f'(x)}}}$或者${\displaystyle {\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}}$

${\displaystyle y=\arcsin x(|x|<1)}$${\displaystyle x=\sin y(\left|y\right|<{\frac {\pi }{2}})}$的反函数，且${\displaystyle x=\sin y}$${\displaystyle I_{y}=\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$开区间上严格单调、可导，且${\displaystyle (\sin y)'=\cos y>0}$因此由反函数求导法则可得：在对应区间${\displaystyle I_{y}=(-1,1)}$内有：

${\displaystyle (\arcsin x)'={\frac {1}{(\sin y)'}}={\frac {1}{\cos y}}={\frac {1}{\sqrt {1-\sin ^{2}y}}}={\frac {1}{\sqrt {1-x^{2}}}}}$

参数方程和极坐标方程的求导

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{dt}}\cdot {\frac {dt}{dx}}={\frac {dy}{dt}}\cdot {\frac {1}{\frac {dx}{dt}}}={\frac {\phi '(t)}{\psi '(t)}}}$

${\displaystyle {\frac {dy}{dx}}={\frac {\left[\rho (\theta )\sin \theta \right]'}{\left[\rho (\theta )\cos \theta \right]'}}={\frac {\rho _{\theta }^{'}\sin \theta +\rho \cos \theta }{\rho _{\theta }^{'}\cos \theta -\rho \sin \theta }}}$

隐函数的求导

• 有关隐函数的定义，参见隐函数

${\displaystyle {\frac {d(x^{\frac {1}{2}}+y^{\frac {1}{2}})}{dx}}={\frac {d{\sqrt {a}}}{dx}}}$

${\displaystyle {\frac {1}{2}}x^{-{\frac {1}{2}}}+{\frac {1}{2}}y^{-{\frac {1}{2}}}\cdot {\frac {dy}{dx}}=0}$

${\displaystyle {\frac {dy}{dx}}=-{\sqrt {\frac {y}{x}}}(x,y>0)}$

• 通过例题，应当注意方程两边求导的对象是${\displaystyle x}$，而${\displaystyle y}$是用${\displaystyle x}$表示的，相当于一个${\displaystyle x}$的复合函数，故根据复合函数的求导法则：${\displaystyle [f(y)]'=f'(y)\cdot y_{x}^{'}}$。本题中${\displaystyle f(y)={\sqrt {y}},f'(y)={\frac {1}{2}}y^{-{\frac {1}{2}}},y_{x}^{'}={\frac {dy}{dx}}}$

高阶导数

${\displaystyle {\frac {{{\rm {d}}^{2}}y}{{\rm {d}}{x^{2}}}}}$ ${\displaystyle ={\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {{\rm {d}}y}{{\rm {d}}x}}\right)}$

${\displaystyle ={\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\phi '(t)}{\psi '(t)}}\right)}$

${\displaystyle ={\frac {\rm {d}}{{\rm {d}}t}}\left({\frac {\phi '(t)}{\psi '(t)}}\right)\cdot {\frac {{\rm {d}}t}{{\rm {d}}x}}}$

${\displaystyle ={\frac {\phi ''(t)\psi '(t)-\phi '(t)\psi ''(t)}{{[\psi '(t)]}^{2}}}\cdot {\frac {1}{\psi '(t)}}}$

基本函数的导数

 基本导数公式 1 ${\displaystyle C'=0}$ 2 ${\displaystyle (x^{\mu })'=\mu x^{\mu -1}}$ 3 ${\displaystyle (\sin x)'=\cos x}$ 4 ${\displaystyle (\cos x)'=-\sin x}$ 5 ${\displaystyle (\tan x)'={\frac {1}{{\cos ^{2}}x}}={\sec ^{2}}x}$ 6 ${\displaystyle (\cot x)'=-{\frac {1}{{\sin ^{2}}x}}=-{\csc ^{2}}x}$ 7 ${\displaystyle (\sec x)'={\sec x}{\tan x}}$ 8 ${\displaystyle (\csc x)'=-{\csc x}{\cot x}}$ 9 ${\displaystyle (\ln x)'={\frac {1}{x}}}$ 10 ${\displaystyle (\log _{a}x)'={\frac {1}{x\ln a}}}$ 11 ${\displaystyle (e^{x})'=e^{x}}$ 12 ${\displaystyle (a^{x})'=a^{x}\ln a}$其中${\displaystyle a>0,a\neq 1}$ 13 ${\displaystyle (\arcsin x)'={\frac {1}{\sqrt {1-x^{2}}}}}$ 14 ${\displaystyle (\arccos x)'=-{\frac {1}{\sqrt {1-x^{2}}}}}$ 15 ${\displaystyle (\arctan x)'={\frac {1}{1+x^{2}}}}$ 16 ${\displaystyle (\operatorname {arccot} x)'=-{\frac {1}{1+x^{2}}}}$