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关于导数的知识整理

常数法则

\[ f(x)=c 且 c 为常数 \]

\[ f'(x)=0 \]

和差法则

\[ f(x)=g(x)+h(x) \]

\[ f'(x)=g'(x)+h'(x) \]

乘积法则

\[ f(x)=g(x) \cdot h(x) \]

\[ f'(x)=g'(x) \cdot h(x)+g(x) \cdot h'(x) \]

商法则

\[ f(x)=\frac{g(x)}{h(x)}且h(x)不为零 \]

\[ f'(x)=\frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2} \]

衍生结论

\[ f(x)=\frac {1}{h(x)} \]

\[ f'(x)=\frac {h'(x)}{h(x)^2} \]

幂法则

\[ f(x)=x^a \]

\[ f'(x)=ax^{a-1} \]

链式法则

\[ 存在f(x)=g(h(x)) \]

\[ f'(x)=g'(h(x)) \cdot g'(x) \]

指数法则

\[ f(x)=a^x \]

\[ f'(x)=a^x \cdot ln(a) \]

对数法则

\[ f(x)=log_a(x) \]

\[ f'(x)=\frac{1}{x \cdot ln(a)} \]

反函数法则

\[ f(x)有反函数f^{-1}(x) \]

\[ f'^{-1}(x)=\frac {1}{f'(x)} \]

三角函数法则

\[ 若f(x)=sin(x)则f'(x)=cos(x) \]
\[ 若f(x)=cos(x)则f'(x)=-sin(x) \]
\[ 若f(x)=tan(x)则f'(x)=sec^2(x) \]
\[ 若f(x)=cot(x)则f'(x)=-csc^2(x) \]
\[ 若f(x)=sec(x)则f'(x)=sec(x)\cdot tan(x) \]
\[ 若f(x)=csc(x)则f'(x)=-csc(x)\cdot cot(x) \]