求导公式总结

基本初等函数的导数公式

(1) $(C)^{\prime}=0$

(2) $\left(x^n\right)^{\prime}=\mu x^{n-1}$,

(3) $(\sin x)^{\prime}=\cos x$

(4) $(\cos x)^{\prime}=-\sin x$,

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

(6) $(\cot x)^{\prime}=-\csc ^2 x$,

(7) $(\sec x)^{\prime}=\sec x \tan x$,

(8) $(\csc x)^{\prime}=-\csc x \cot x$

(9) $\left(a^x\right)^{\prime}=a^x \ln a$

(10) $\left(\mathrm{e}^{x}\right)^{\prime}=\mathrm{e}^x$,

(11) $\left(\log _a x\right)^{\prime}=\frac{1}{x \ln a}$,

(12) $(\ln x)^{\prime}=\frac{1}{x}$,

(13) $(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^2}}$,

(14) $(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^2}}$,

(15) $(\arctan x)^{\prime}=\frac{1}{1+x^2}$,

(16) $(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^2}$.

函数的和差积商求导法则

$(u \pm v)^{\prime}=u^{\prime} \pm v^{\prime}$

$(C u)^{\prime}=C u^{\prime}$

$(u v)^{\prime}=u^{\prime} v+u v^{\prime}$

$\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^2} \quad(v \neq 0)$

复合函数求导法则

$(f(g(x)))^{\prime}=f^{\prime}(g(x)) \cdot g^{\prime}(x)$

隐函数求导法则

$f(x, y)=0 \Rightarrow y^{\prime}=-\frac{f_x(x, y)}{f_y(x, y)}$

参数方程求导法则

$\left\{\begin{array}{l}x=\varphi(t) \\ y=\psi(t)\end{array}, y^{\prime}=\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right.$

反函数求导法则

$$ \left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)} $$

变限积分求导法则

$$ \begin{aligned} & {\left[\int_a^x f(t) d t\right]^{\prime}=f(x)} \\ & {\left[\int_x^b f(t) d t\right]^{\prime}=-f(x)} \\ & {\left[\int_a^{u(x)} f(t) d t\right]^{\prime}=f(u(x)) u^{\prime}(x)} \\ & {\left[\int_{v(x)}^{u(x)} f(t) d t\right]^{\prime}=f(u(x)) u^{\prime}(x)-f(v(x)) v^{\prime}(x)} \end{aligned} $$

定义计算导数

$f^{\prime}\left(x_0\right)=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{f\left(x_0+\Delta x\right)-f\left(x_0\right)}{\Delta x}$,