关于导数的知识整理¶
常数法则¶
若
\[
f(x)=c 且 c 为常数
\]
则
\[
f'(x)=0
\]
和差法则¶
若
\[
f(x)=g(x)+h(x)
\]
则
\[
f'(x)=g'(x)+h'(x)
\]
乘积法则¶
若
\[
f(x)=g(x) \cdot h(x)
\]
则
\[
f'(x)=g'(x) \cdot h(x)+g(x) \cdot h'(x)
\]
商法则¶
若
\[
f(x)=\frac{g(x)}{h(x)}且h(x)不为零
\]
则
\[
f'(x)=\frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2}
\]
衍生结论¶
若
\[
f(x)=\frac {1}{h(x)}
\]
则
\[
f'(x)=\frac {h'(x)}{h(x)^2}
\]
幂法则¶
若
\[
f(x)=x^a
\]
则
\[
f'(x)=ax^{a-1}
\]
链式法则¶
若
\[
存在f(x)=g(h(x))
\]
则
\[
f'(x)=g'(h(x)) \cdot g'(x)
\]
指数法则¶
若
\[
f(x)=a^x
\]
则
\[
f'(x)=a^x \cdot ln(a)
\]
对数法则¶
若
\[
f(x)=log_a(x)
\]
则
\[
f'(x)=\frac{1}{x \cdot ln(a)}
\]
反函数法则¶
若
\[
f(x)有反函数f^{-1}(x)
\]
则
\[
f'^{-1}(x)=\frac {1}{f'(x)}
\]
三角函数法则¶
\[
若f(x)=sin(x)则f'(x)=cos(x)
\]
\[
若f(x)=cos(x)则f'(x)=-sin(x)
\]
\[
若f(x)=tan(x)则f'(x)=sec^2(x)
\]
\[
若f(x)=cot(x)则f'(x)=-csc^2(x)
\]
\[
若f(x)=sec(x)则f'(x)=sec(x)\cdot tan(x)
\]
\[
若f(x)=csc(x)则f'(x)=-csc(x)\cdot cot(x)
\]