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导数公式

admin 12月 06, 2020 0

基本初等函数求导公式

$(C)' = 0$
$(x^u)'=ux^{u-1}$
$(sin x)' = cos x $
$(tan x)' = sec^2 x $
$(cos x)' = -sin x $
$(cot x)' = - csc^2 x$
$(sec x)' =sec x * tan x$
$(csc x)' = -csc x * cot x$
$(a^x)' = a^x ln a$
$(e^x)' = e^x$
$(log_a x)' = {1 over xln a} $
$(ln x)' = {1 over x}$
$(arcsin x)' = {frac1{sqrt{1-x^2}} }$
$(arccos x)' = {-frac1{sqrt{1-x^2}} }$
$(arctan x)' = {frac1{1+x^2} }$
$(arctan x)' = {-frac1{1+x^2} }$

函数求导法则

设 $z =z(x),f = f(x)$ 都可导，则：
1.$(u{pm}f)'=u'pm v'$
2.$(uv)'=u'v + uv'$
3.$(frac{u}{v})' = frac {u'v-uv'}{v^2}$

链式法则

设 $z =z(x),f = f(z)$ 都可导，则:
$frac{partial f}{partial x} = frac{partial f}{partial z} * frac{partial z}{partial x}$

矩阵转置求导(前导不变，后导转置)

$(AX)' = A^T$
$(XA)' = A$
$(x^TA)' = A$
$(Ax^T)' = A^T$